AP Physics C Calculus Formulas: All 11 Printed Rules

The AP Physics C booklet prints eleven calculus rules in one box: six derivatives, including the chain rule, and five integrals. Neither AP Physics 1 nor AP Physics 2 prints any calculus. There is no product rule, no quotient rule, no integration by parts and no constant of integration.

The CALCULUS box is printed only in the two AP Physics C Course and Exam Descriptions effective Fall 2024: C: Mechanics appendix page 206 and C: Electricity and Magnetism appendix page 181. The two printings are identical. AP Physics 1 and AP Physics 2 print no calculus.

The Box as the Booklet Prints It

The box is headed CALCULUS and sits in the middle of three boxes on the final appendix page of each Physics C booklet, between VECTORS on its left and IDENTITIES on its right. It contains eleven rules: six derivatives, then five integrals, in that order, with no subheadings separating them.

Derivatives

Printed rule	Common name
$\dfrac{df}{dx} = \dfrac{df}{du}\dfrac{du}{dx}$	Chain rule
$\dfrac{d}{dx}\left(x^n\right) = nx^{n-1}$	Power rule
$\dfrac{d}{dx}\left(e^{ax}\right) = ae^{ax}$	Exponential
$\dfrac{d}{dx}\left(\ln ax\right) = \dfrac{1}{x}$	Natural logarithm
$\dfrac{d}{dx}\left[\sin(ax)\right] = a\cos(ax)$	Sine
$\dfrac{d}{dx}\left[\cos(ax)\right] = -a\sin(ax)$	Cosine

Integrals

Printed rule	Common name
$\displaystyle\int x^n\,dx = \dfrac{1}{n+1}x^{n+1}$ , with $n \neq -1$	Power rule
$\displaystyle\int e^{ax}\,dx = \dfrac{1}{a}e^{ax}$	Exponential
$\displaystyle\int \dfrac{dx}{x+a} = \ln\lvert x+a\rvert$	Logarithmic
$\displaystyle\int \cos(ax)\,dx = \dfrac{1}{a}\sin(ax)$	Cosine
$\displaystyle\int \sin(ax)\,dx = -\dfrac{1}{a}\cos(ax)$	Sine

Eleven rules in total: six above, five below. The restriction $n \neq -1$ is printed on the integral power rule, and the absolute-value bars are printed on the logarithmic integral. Neither is something the sheet leaves to you.

Where it is printed:

AP Physics C: Mechanics, appendix page 206
AP Physics C: Electricity and Magnetism, appendix page 181

The two printings are identical. Neither algebra-based booklet prints any calculus at all, which is the cleanest single statement of what separates the two course families on paper.

The Constant Pattern, Which Is the Fastest Thing to Learn

Look down the two lists and one structure repeats in six of the eleven rules. A constant $a$ sits inside the function, and it comes out multiplying on differentiation and dividing on integration.

Operation	What happens to the inner constant
$\dfrac{d}{dx}e^{ax}$	multiply by $a$
$\displaystyle\int e^{ax}\,dx$	divide by $a$
$\dfrac{d}{dx}\sin(ax)$	multiply by $a$
$\displaystyle\int \cos(ax)\,dx$	divide by $a$
$\dfrac{d}{dx}\cos(ax)$	multiply by $a$ , and change sign
$\displaystyle\int \sin(ax)\,dx$	divide by $a$ , and change sign

That is the chain rule showing through, and it is where marks are actually lost. Differentiating $x = x_{max}\cos(\omega t + \phi)$ produces $-\omega x_{max}\sin(\omega t + \phi)$ , and dropping the $\omega$ leaves you with a velocity amplitude that is wrong by a factor of the angular frequency. Differentiating $e^{-t/RC}$ produces a factor of $-1/RC$ , and dropping it leaves a current with the wrong magnitude and the wrong sign.

The sign rule is simpler than it looks. Only two of the eleven rules carry a minus sign, and they are the cosine derivative and the sine integral. Everything else is positive.

What Each Derivative Rule Is Printed For

A rule you cannot attach to a physics result is a rule you will not reach for under time pressure. Each of the six derivatives is on the sheet because a specific AP Physics C result depends on it.

Printed derivative	The AP result it produces
$\dfrac{df}{dx} = \dfrac{df}{du}\dfrac{du}{dx}$	Every differentiation with a compound argument on the sheet. It is what brings the $\omega$ out of $\cos(\omega t + \phi)$ and the $-1/RC$ out of $e^{-t/RC}$ . It also licenses the substitution $a = \dfrac{dv}{dt} = v\dfrac{dv}{dx}$ , which converts a force that depends on position into an energy statement without ever solving for $v(t)$ .
$\dfrac{d}{dx}(x^n) = nx^{n-1}$	Velocity and acceleration from a polynomial position function, $v_x = dx/dt$ and $a_x = dv_x/dt$ . Force from potential energy, printed as $F_x = -\dfrac{dU(x)}{dx}$ : applied to $U_s = \frac{1}{2}k(\Delta x)^2$ it returns $F_s = -k\Delta x$ , and applied to $U_G = -Gm_1m_2/r$ it returns the inverse-square gravitational force. The same rule turns $V$ into $E_x = -dV/dx$ on the E&M sheet.
$\dfrac{d}{dx}(e^{ax}) = ae^{ax}$	The current in a charging or discharging RC circuit, from $I = dq/dt$ . The back emf in an LR circuit, from $\varepsilon = -L\,dI/dt$ . The acceleration of an object slowing under a resistive force proportional to speed, on its way to terminal velocity.
$\dfrac{d}{dx}(\ln ax) = \dfrac{1}{x}$	The field of an infinite line of charge recovered from its potential. Because $V$ varies as $\ln r$ for a line, $E_r = -dV/dr$ varies as $1/r$ , which is the field Gauss's law gives independently.
$\dfrac{d}{dx}[\sin(ax)] = a\cos(ax)$	Velocity from a sine-form SHM displacement, and the induced emf from a loop whose flux varies sinusoidally as it rotates.
$\dfrac{d}{dx}[\cos(ax)] = -a\sin(ax)$	Applied twice to the printed $x = x_{max}\cos(\omega t + \phi)$ , it gives $v = -\omega x_{max}\sin(\omega t + \phi)$ and then $a = -\omega^2 x_{max}\cos(\omega t + \phi) = -\omega^2 x$ . That last equality is the definition of simple harmonic motion, produced entirely by this one rule.

The last row is worth pausing on. The AP definition of SHM is that acceleration is proportional to displacement and directed opposite to it. The sheet does not print $a = -\omega^2 x$ . It prints the position function and it prints the cosine derivative, and it expects you to connect them.

What Each Integral Rule Is Printed For

The integrals are the higher-value half of the box, because the C sheets state a dozen quantities as integrals and give you no closed forms for them.

Printed integral	The AP result it produces
$\displaystyle\int x^n\,dx = \dfrac{1}{n+1}x^{n+1}$	Work from a variable force, $W = \int \vec{F}\cdot d\vec{r}$ , including the spring case that integrates to $\frac{1}{2}k(\Delta x)^2$ . Rotational inertia of a continuous body, $I = \int r^2\,dm$ . Displacement and velocity changes, $\Delta x = \int v_x\,dt$ and $\Delta v_x = \int a_x\,dt$ . Enclosed charge in a non-uniform distribution, $Q = \int \rho(r)\,dV$ .
$\displaystyle\int e^{ax}\,dx = \dfrac{1}{a}e^{ax}$	The total charge delivered by a decaying current, $\Delta q = \int I\,dt$ , which is how you show a discharging capacitor delivers exactly $q_0$ over infinite time. Also the impulse of an exponentially decaying force.
$\displaystyle\int \dfrac{dx}{x+a} = \ln\lvert x+a\rvert$	Solving the separable differential equations that generate every exponential on the sheet. $dq/dt = -q/(RC)$ separates to $\int dq/q = -\int dt/(RC)$ , integrates to a logarithm, and exponentiates to $q = q_0 e^{-t/RC}$ . The same route gives the LR current and the approach to terminal velocity. It also gives the potential difference near an infinite line charge, which goes as $\ln(r_2/r_1)$ .
$\displaystyle\int \cos(ax)\,dx = \dfrac{1}{a}\sin(ax)$	Displacement recovered from an SHM velocity, and the work done by a force that varies sinusoidally with position.
$\displaystyle\int \sin(ax)\,dx = -\dfrac{1}{a}\cos(ax)$	The same family, and the average of a sinusoidal quantity over part of a cycle.

Notice the division of labour between the first and the third rule. The power rule handles every exponent except $-1$ , and it says so in print. The case it excludes is exactly the case the logarithmic integral covers. The two lines are written to tile the space between them, which is why $\int dx/x$ in bare form is not printed separately: set $a = 0$ in the third rule and you have it.

The Power Rule for Integration Is the Whole Course in One Line

If you take one rule out of this box, take $\int x^n\,dx = \frac{1}{n+1}x^{n+1}$ . Two of the most heavily weighted results in AP Physics C are direct applications of it, and both of them are printed on the sheet as integrals with nothing evaluated.

Work from a variable force. The sheet prints $W = \int_a^b \vec{F}\cdot d\vec{r}$ and stops. For a spring the force magnitude is $kx$ , so the work you do stretching it from 0 to $x$ is

W = \int_0^x kx'\,dx' = k\left[\frac{x'^2}{2}\right]_0^x = \frac{1}{2}kx^2

which is the elastic potential energy the sheet also prints, as $U_s = \frac{1}{2}k(\Delta x)^2$ . The power rule with $n = 1$ is the only step. That is the calculus-based derivation of a result Physics 1 students are asked to memorise.

Rotational inertia of a continuous body. The sheet prints $I = \int r^2\,dm$ and stops, and it prints no rotational inertia for any actual shape. For a uniform rod of mass $M$ and length $L$ rotating about one end, write $dm = \lambda\,dr$ with $\lambda = M/L$ , which is the linear mass density the sheet defines as $\lambda = \frac{d}{d\ell}m(\ell)$ . Then

I = \int_0^L r^2 \frac{M}{L}\,dr = \frac{M}{L}\left[\frac{r^3}{3}\right]_0^L = \frac{1}{3}ML^2

The power rule with $n = 2$ is again the only step. Change the limits to $-L/2$ and $L/2$ and the same integral gives $\frac{1}{12}ML^2$ for rotation about the centre, which is also the parallel-axis theorem checking out: $\frac{1}{12}ML^2 + M(L/2)^2 = \frac{1}{3}ML^2$ .

Those two derivations are why the College Board can decline to print a table of moments of inertia. It expects you to build one. The rotational inertia topic page covers the AP framing of that expectation.

The Chain Rule Line, and the Substitution It Licenses

The chain rule is printed in Leibniz form, $\frac{df}{dx} = \frac{df}{du}\frac{du}{dx}$ , rather than as $[f(g(x))]' = f'(g(x))g'(x)$ . The Leibniz form is the useful one in physics, because it looks like fractions cancelling and that is exactly how it gets used.

The move worth knowing is the velocity substitution. Acceleration is $a = dv/dt$ , but for a force that depends on position rather than time, that is the wrong variable. Insert $dx$ on top and bottom:

a = \frac{dv}{dt} = \frac{dv}{dx}\frac{dx}{dt} = v\frac{dv}{dx}

Now Newton's second law for a position-dependent force reads $F(x) = mv\,dv/dx$ , which separates into $F(x)\,dx = mv\,dv$ and integrates on both sides to give $\int F\,dx = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$ . That is the work-energy theorem, derived rather than assumed, using the chain rule and the integral power rule and nothing else.

The same substitution is the standard route through a resistive-force problem, which is a C: Mechanics topic the algebra-based courses never see. If the drag force depends on speed, you separate variables and integrate, and the chain rule is what lets you choose whether to integrate with respect to time or with respect to position depending on which the question asks for.

Why the Logarithm Derivative Has No a in It

$\frac{d}{dx}(\ln ax) = \frac{1}{x}$ is the line students most often assume is a misprint. Every other rule in the box that has an $a$ inside the function produces an $a$ in the answer. This one does not, and the printed form is correct.

The reason is one line of algebra. By the logarithm identity printed in the box next door, $\ln(ax) = \ln a + \ln x$ . The term $\ln a$ is a constant, so it differentiates to zero, and what is left is $\frac{d}{dx}\ln x = 1/x$ . The constant scale factor genuinely does not survive.

That is a useful physics fact, not just a curiosity. It means the slope of a log-linear graph does not depend on the units you measured in. If you plot $\ln q$ against $t$ for a discharging capacitor, switching from coulombs to microcoulombs shifts every point vertically by the constant $\ln(10^6)$ and leaves the slope, and therefore the extracted time constant, untouched. Changing units cannot change a physical result, and this rule is the mathematical reason it cannot.

One genuine gap: the sheet prints no rule for $\ln(ax + b)$ or for a logarithm to any base other than $e$ . If you need those, the identity box next door gets you there.

What the Box Does Not Print

The eleven rules above are the whole box. None of the following appears anywhere in either Physics C booklet, and knowing that in advance stops you hunting for it in an exam:

The product rule and the quotient rule. Neither is printed, and both are assumed. You will need the product rule for any flux that changes because both the field and the area are changing.
Integration by parts. Not printed and, in practice, not required: AP Physics C integrals are chosen so that the printed rules plus a substitution suffice.
The constant of integration. All five integrals are written without $+ C$ . On a free-response question you are expected to supply it, or to use limits, and to fix it from an initial condition. A solution that quietly drops an initial value will lose the point even though the integral matches the sheet.
Limits of integration. All five printed integrals are indefinite. Several of the physics equations on the same sheet are written with limits, such as $W = \int_a^b \vec{F}\cdot d\vec{r}$ and $\vec{J} = \int_{t_1}^{t_2}\vec{F}_{net}(t)\,dt$ , so you must move between the two forms yourself.
Derivatives or integrals of tangent, secant, cosecant or cotangent. None of those four functions appears anywhere in any AP Physics booklet.
Inverse trigonometric results, so no $\int dx/(1 + x^2) = \arctan x$ and no $\int dx/\sqrt{a^2 - x^2}$ .
$\frac{d}{dx}(a^x)$ for a general base. Only the base $e$ is printed.
Partial derivatives, the del operator, divergence or curl. Maxwell's equations appear on the C: E&M sheet in integral form only, and the differential forms are not printed and are not examinable.
Taylor series or the binomial approximation $(1 + x)^n \approx 1 + nx$ , which is the standard tool for showing that a dipole field falls off as $1/r^3$ far away.
The fundamental theorem of calculus stated as such, and no statement that a derivative is the slope of a tangent or that an integral is the area under a curve. Those conceptual bridges are assumed, and they are what AP graph-reading questions test.

A caution about verifying any of this elsewhere. Transcriptions of the AP equation sheet, including this site's own machine-readable copy, cover the physics equations and the constants boxes and deliberately omit the geometry, trigonometry, vector, calculus and identity tables. A calculus rule being absent from such a transcription tells you nothing about the booklet. Every rule on this page was read from a rendered image of appendix page 206 of the C: Mechanics CED and appendix page 181 of the C: E&M CED, both Effective Fall 2024.

Physics 1 and Physics 2 Candidates

There is no calculus box in either algebra-based booklet, and there is no calculus anywhere in either algebra-based equation set. AP Physics 1 fits its whole reference onto appendix pages 211 and 212, and AP Physics 2 onto 218 through 221, and no derivative or integral appears on any of them.

That is a real difference in what the exams can ask, and it shows up as a difference in what is printed. Compare the same three quantities across the courses:

Quantity	Algebra-based printed form	Physics C printed form
Net force	$\vec{F}_{net} = \dfrac{\Delta \vec{p}}{\Delta t} = m\dfrac{\Delta \vec{v}}{\Delta t} = m\vec{a}$	$\vec{F}_{net} = \dfrac{d\vec{p}}{dt}$
Impulse	$\vec{J} = \vec{F}_{avg}\Delta t = \Delta\vec{p}$	$\vec{J} = \displaystyle\int_{t_1}^{t_2}\vec{F}_{net}(t)\,dt = \Delta\vec{p}$
Work	$W = F_{\parallel}d = Fd\cos\theta$	$W = \displaystyle\int_a^b \vec{F}\cdot d\vec{r}$

Same physics, two levels of generality. The algebra-based forms assume the quantity is constant, or ask you to use an average; the Physics C forms do not, which is why the C exams can set a force that varies with position and the Physics 1 exam cannot without giving you a graph to read the area from.

If you are moving from Physics 1 to Physics C, the calculus box is the map between the two columns above. Every algebra-based formula with a $\Delta$ in it becomes a derivative, and every algebra-based formula that multiplies a constant by an interval becomes an integral.

Rotational inertia of a rod, built from the printed power rule

A uniform thin rod has mass M = 2.0 kg and length L = 1.2 m. Using only the relations printed on the C: Mechanics sheet, find its rotational inertia about an axis through one end and perpendicular to the rod.

The sheet prints the definition $I = \int r^2\,dm$ and no closed form for any shape, so the integral has to be done.
Express $dm$ in terms of position. The sheet defines linear mass density as $\lambda = \frac{d}{d\ell}m(\ell)$ , so for a uniform rod $\lambda = M/L = 2.0/1.2 = 1.6667$ kg/m and $dm = \lambda\,dr$ .
Set the origin at the axis, so $r$ runs from 0 to L: $I = \int_0^L r^2 \lambda\,dr = \lambda\int_0^L r^2\,dr$ .
Apply the printed integral power rule with $n = 2$ : $\int r^2\,dr = \frac{1}{3}r^3$ , so $I = \lambda\left[\frac{r^3}{3}\right]_0^L = \frac{\lambda L^3}{3}$ .
Substitute $\lambda = M/L$ to get the general result: $I = \frac{M L^2}{3}$ .
Numbers: $I = \frac{(2.0)(1.2)^2}{3} = \frac{(2.0)(1.44)}{3} = \frac{2.88}{3} = 0.96$ kg m $^2$ .
Check with the parallel-axis theorem, also printed. About the centre the same integral from $-L/2$ to $L/2$ gives $I_{cm} = \frac{1}{12}ML^2 = \frac{(2.0)(1.44)}{12} = 0.24$ kg m $^2$ . Shifting to the end: $I' = I_{cm} + Md^2 = 0.24 + (2.0)(0.60)^2 = 0.24 + 0.72 = 0.96$ kg m $^2$ . The two routes agree.

$I = \frac{1}{3}ML^2 = 0.96$ kg m $^2$ about the end. The only calculus used was the printed power rule with n = 2, and the parallel-axis theorem confirms it independently.

Current in a charging RC circuit, from the printed exponential derivative

A 12 V ideal battery charges a 4.0 microfarad capacitor through a 5.0 kilohm resistor. The charge on the capacitor is $q(t) = Q_{max}\left(1 - e^{-t/RC}\right)$ . Find the current at t = 0 and at t = 0.020 s.

Time constant first: $\tau = RC = (5.0 \times 10^3)(4.0 \times 10^{-6}) = 0.020$ s. The E&M sheet prints $\tau = R_{eq}C_{eq}$ .
Final charge: $Q_{max} = C\varepsilon = (4.0 \times 10^{-6})(12) = 4.8 \times 10^{-5}$ C.
Current is $I = dq/dt$ , which the sheet prints. Differentiate the given $q(t)$ term by term. The constant $Q_{max}$ differentiates to zero, and the printed rule $\frac{d}{dx}(e^{ax}) = ae^{ax}$ handles the rest with $a = -1/RC$ : $\frac{d}{dt}\left(-e^{-t/RC}\right) = +\frac{1}{RC}e^{-t/RC}$ .
So $I = \frac{Q_{max}}{RC}e^{-t/RC}$ . Note the factor of $1/RC$ : it came entirely from the constant inside the exponent, and forgetting it is the standard error here.
At $t = 0$ : $I_0 = \frac{4.8 \times 10^{-5}}{0.020} = 2.4 \times 10^{-3}$ A. Sanity check against the uncharged-capacitor shortcut $I_0 = \varepsilon/R = 12/(5.0 \times 10^3) = 2.4 \times 10^{-3}$ A. They agree, which confirms the missing factor was not missed.
At $t = 0.020$ s, which is exactly one time constant: $I = (2.4 \times 10^{-3})e^{-1} = (2.4 \times 10^{-3})(0.36788) = 8.83 \times 10^{-4}$ A.

$I_0 = 2.4 \times 10^{-3}$ A and $I(0.020\ \text{s}) = 8.8 \times 10^{-4}$ A. The current has fallen to $1/e$ , about 37 percent, of its initial value after one time constant, which is what the time constant means.

Frequently asked questions

What calculus formulas are on the AP Physics C equation sheet?

Eleven, in a box headed CALCULUS. Six derivatives: the chain rule, the power rule, the derivative of e to the ax, the derivative of ln ax, and the derivatives of sine and cosine of ax. Five integrals: the power rule with the restriction n not equal to minus one, the integral of e to the ax, the integral of dx over x plus a giving a natural logarithm, and the integrals of cosine and sine of ax. They are printed on appendix page 206 of the C: Mechanics booklet and page 181 of the C: Electricity and Magnetism booklet.

Is the product rule on the AP Physics C formula sheet?

No. Neither the product rule nor the quotient rule is printed anywhere in either Physics C booklet, and neither is integration by parts. The only derivative rule the sheet gives for combining functions is the chain rule, written in Leibniz form. The product rule is still assumed knowledge, and you will need it for any flux whose field and area both change with time.

Does the AP Physics C sheet include the constant of integration?

No. All five printed integrals are indefinite and none carries a plus C. You are still expected to include it, or to use limits of integration, and to fix the constant from an initial condition. Several physics equations on the same sheet are written with limits, such as work as the integral of force over displacement, so you have to convert between the indefinite printed rules and the definite integrals the physics needs.

Is calculus on the AP Physics 1 formula sheet?

No. Neither AP Physics 1 nor AP Physics 2 prints any calculus, and no derivative or integral appears in either algebra-based equation set. Their booklets share the geometry box and the trig values table with Physics C but not the vectors, calculus or identities boxes. Physics 1 fits its whole reference on appendix pages 211 and 212, and Physics 2 on pages 218 through 221.

Why does the derivative of ln ax not have an a in it?

Because ln of ax equals ln a plus ln x, and ln a is a constant that differentiates to zero. What remains is the derivative of ln x, which is one over x. The printed form is correct. It also has a physical meaning: the slope of a logarithmic graph does not depend on the units you plotted in, which is why extracting a time constant from a log-linear plot gives the same answer in coulombs or microcoulombs.

Are moments of inertia given on the AP Physics C sheet?

No. The sheet prints only the definitions, I equals the sum of m times r squared and I equals the integral of r squared dm, plus the parallel-axis theorem. No formula for a rod, disc, hoop, sphere or shell appears. That is precisely why the integral power rule is printed: a uniform rod about its end integrates to one third M L squared in two lines, and about its centre to one twelfth M L squared.

Which single calculus rule matters most for AP Physics C?

The integral power rule, the integral of x to the n equals x to the n plus one over n plus one. It is what turns the sheet's work integral into one half k x squared for a spring and its rotational-inertia integral into a closed form for any uniform body, and it also handles displacement from a velocity function and enclosed charge from a charge density. The exponential rules matter almost as much, because every circuit transient on the E&M sheet depends on them.

Do I need to know derivatives of tan and sec for AP Physics C?

They are not printed. Only sine, cosine, the exponential, the natural logarithm and the power rule appear in the calculus box, and tangent, secant, cosecant and cotangent do not appear anywhere in any AP Physics booklet. Nor are inverse trigonometric integrals printed. AP Physics C integrals are constructed so that the eleven printed rules, plus a substitution, are enough.