AP Calculus AB and BC

Derivative Rules: The Complete Table

The core rules are constant (derivative 0), power (x^n to n x^(n-1)), constant multiple, sum, difference, product, quotient, and chain. On top are the six trig derivatives, the inverse trig derivatives, and the exponential and log rules. Pick a rule by the function's outermost operation.

How to pick the right rule

Choosing a derivative rule is a recognition task, not a guessing game. Look at the outermost operation: the last thing you would do if you plugged in a number. That single question routes you to the right rule almost every time. The table below maps what a function looks like to the rule it needs, and the sections after it give the formulas grouped so the patterns are easy to remember.

The one question that decides

Identify the outermost operation. A sum wants the sum rule, a product wants the product rule, a composition wants the chain rule. Only after that do you differentiate the inside pieces.

When the function looks like	Use this rule	Why / watch for
A number alone, like $7$ or $\pi$	Constant	A constant never changes, so its slope is $0$ .
A single power, like $x^5$ or $x^{-2}$	Power	Works for any real exponent, including negatives and roots.
Terms joined by $+$ or $-$	Sum and difference	Differentiate each term on its own, then recombine.
A number times a function, like $4\cos x$	Constant multiple	The coefficient rides along: the derivative is $-4\sin x$ .
Two functions multiplied	Product	Not $f'g'$ : each factor pairs with the other's derivative.
One function over another	Quotient	First check whether a rewrite turns it into a sum of powers.
A function nested inside another, like $\sin(x^2)$	Chain	Outside derivative times inside derivative; keep the inside.

These are Topics 2.5, 2.6, 2.8, 2.9, and 3.1 in the CED. The trig, inverse trig, exponential, and log rows below plug into these structural rules: whenever an inside function is more than a bare $x$ , the chain rule multiplies by that inside function's derivative.

Structural rules (Topics 2.5, 2.6, 2.8, 2.9, 3.1)

These seven rules combine or transform other functions. They are the scaffolding: the specific function derivatives in the later tables get fed into them.

Rule	Formula	When to use it
Constant	$\dfrac{d}{dx}[c] = 0$	A number standing alone. Its rate of change is zero.
Power	$\dfrac{d}{dx}[x^{n}] = n x^{n-1}$	Any single power of $x$ , including negative and fractional exponents. The workhorse for polynomials and radicals.
Constant multiple	$\dfrac{d}{dx}[c\cdot f(x)] = c\cdot f'(x)$	A numeric coefficient on a function. Differentiate the function and keep the coefficient out front.
Sum and difference	$\dfrac{d}{dx}[f \pm g] = f' \pm g'$	A function written as terms joined by $+$ or $-$ . Handle each term separately.
Product	$\dfrac{d}{dx}[fg] = f'g + fg'$	Two nonconstant functions multiplied. The two terms are added, so their order is free.
Quotient	$\dfrac{d}{dx}\left[\dfrac{f}{g}\right] = \dfrac{g f' - f g'}{g^{2}}$	One function divided by another when the bottom is not a single power. Order is fixed: the $g f'$ term comes first.
Chain	$\dfrac{d}{dx}\big[f(g(x))\big] = f'(g(x)) \cdot g'(x)$	A composition, one function inside another. Multiply the outside derivative by the inside derivative.

The two most common slips

The product rule is $f'g + fg'$ , never $f'g'$ . The quotient rule subtracts in a fixed order, and reversing the two numerator terms flips the sign of the whole answer.

The six trigonometric derivatives (Topics 2.7 and 2.10)

Sine and cosine are the two you memorize outright (Topic 2.7). The other four come from rewriting tangent, cotangent, secant, and cosecant with identities and differentiating (Topic 2.10). All six formulas below assume the angle is in radians.

Function	Derivative	Why / when
$\sin x$	$\cos x$	Memorize directly; the anchor for the rest.
$\cos x$	$-\sin x$	A co-function, so its derivative carries a minus sign.
$\tan x$	$\sec^{2} x$	From the quotient rule on $\sin x / \cos x$ .
$\cot x$	$-\csc^{2} x$	Co-function of tangent; note the minus sign.
$\sec x$	$\sec x \tan x$	From the chain rule on $(\cos x)^{-1}$ .
$\csc x$	$-\csc x \cot x$	Co-function of secant; carries a minus sign.

Two patterns that cut the memorizing in half

Every co-function (cosine, cotangent, cosecant) has a derivative that carries a minus sign. And when the angle is a function of $x$ rather than a bare $x$ , the chain rule tacks on the inside derivative: $\dfrac{d}{dx}[\sin u] = \cos u \cdot u'$ .

Inverse trig, exponential, and log derivatives (Topics 2.7 and 3.4)

The exponential and log rules for $e^{x}$ and $\ln x$ are Topic 2.7; the general bases $a^{x}$ and $\log_a x$ come from them by rewriting. Notice how $\ln a$ shows up as an extra factor whenever the base is not $e$ .

Function	Derivative	Why / when
$e^{x}$	$e^{x}$	Equal to its own derivative: every solution of $f' = f$ is $Ce^{x}$ .
$a^{x}$	$a^{x} \ln a$	General base: the extra factor is $\ln a$ . When $a = e$ , $\ln e = 1$ recovers $e^{x}$ .
$\ln x$	$\dfrac{1}{x}$	For $x > 0$ . Turns a logarithm into a simple power.
$\log_a x$	$\dfrac{1}{x \ln a}$	Change of base puts $\ln a$ in the denominator.

The inverse trig derivatives (Topic 3.4) come from the inverse-function rule and the Pythagorean identities. AB and BC most often test $\arcsin$ , $\arccos$ , and $\arctan$ ; the other three are listed for completeness. The co-function pattern repeats: each co-inverse has the same size derivative as its partner but the opposite sign.

Function	Derivative	Why / when
$\arcsin x$	$\dfrac{1}{\sqrt{1-x^{2}}}$	Domain $\|x\| < 1$ . The inverse trig derivative AP uses most.
$\arccos x$	$-\dfrac{1}{\sqrt{1-x^{2}}}$	Same size as $\arcsin$ , opposite sign (co-function).
$\arctan x$	$\dfrac{1}{1+x^{2}}$	Defined for all $x$ ; no radical, no sign trap.
$\operatorname{arccot} x$	$-\dfrac{1}{1+x^{2}}$	Co-function of $\arctan$ ; opposite sign.
$\operatorname{arcsec} x$	$\dfrac{1}{\|x\|\sqrt{x^{2}-1}}$	Domain $\|x\| > 1$ ; keep the absolute value: dropping it flips the sign for $x < -1$ .
$\operatorname{arccsc} x$	$-\dfrac{1}{\|x\|\sqrt{x^{2}-1}}$	Co-function of $\operatorname{arcsec}$ ; opposite sign.

There is no formula sheet

The AP Calculus AB and BC exams provide no reference sheet, so every rule on this page has to be memorized or rebuilt quickly. With a chain rule, each inside function $u$ contributes a factor of $u'$ : for example $\dfrac{d}{dx}[\arctan u] = \dfrac{u'}{1+u^{2}}$ .