AP Calculus AB — Unit 3 Formula Sheet

Vocabulary and formula review flashcards covering AP Calculus AB Unit 3 topics: basic differentiation rules, product and quotient rules, chain rule, transcendental derivatives, implicit differentiation, inverse functions, and radicals.

Power Rule

The differentiation rule stating that $\frac{d}{dx}(x^n) = n(x^{n-1})$.

Constant Rule

The differentiation rule stating that $\frac{d}{dx}(c) = 0$.

Constant Multiple Rule

The differentiation rule stating that $(cf)' = cf'$.

Sum Rule

The differentiation rule stating that $(f + g)' = f' + g'$.

Difference Rule

The differentiation rule stating that $(f - g)' = f' - g'$.

Product Rule

A rule used when two functions are multiplied, given by $(fg)' = f'g + fg'$.

Quotient Rule

A rule used when one function is divided by another, given by $(f/g)' = \frac{gf' - fg'}{g^2}$, remembered as bottom times derivative of top, minus top times derivative of bottom, divided by bottom squared.

Chain Rule

A rule used when one function is inside another function, defined as $(f(g(x)))' = f'(g(x)) \times g'(x)$.

Derivative of $e^x$

$$e^x$$

Derivative of $a^x$

$$a^x \times \ln(a)$$

Derivative of $\ln(x)$

$$\frac{1}{x}$$

Derivative of $\log_a(x)$

$$\frac{1}{x \times \ln(a)}$$

Derivative of $\ln(u)$ with Chain Rule

$$\frac{u'}{u}$$

Derivative of $e^u$ with Chain Rule

$$e^u \times u'$$

Trigonometric Derivatives: $\sin(x)$ and $\cos(x)$

The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$.

Trigonometric Derivatives: $\tan(x)$ and $\cot(x)$

The derivative of $\tan(x)$ is $\sec^2(x)$, and the derivative of $\cot(x)$ is $-\csc^2(x)$.

Trigonometric Derivatives: $\sec(x)$ and $\csc(x)$

The derivative of $\sec(x)$ is $\sec(x)\tan(x)$, and the derivative of $\csc(x)$ is $-\csc(x)\cot(x)$.

Derivative of $\arcsin(x)$

$$\frac{1}{\sqrt{1 - x^2}}$$

Derivative of $\arccos(x)$

$$-\frac{1}{\sqrt{1 - x^2}}$$

Derivative of $\arctan(x)$

$$\frac{1}{1 + x^2}$$

Derivative of $\arcsin(u)$ with Chain Rule

$$\frac{u'}{\sqrt{1 - u^2}}$$

Implicit Differentiation

A technique used when $x$ and $y$ appear together in an equation; whenever you differentiate a term containing $y$, multiply by $\frac{dy}{dx}$.

Derivatives of Inverse Functions

If $f$ and $g$ are inverse functions: $(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}$; if $f(a) = b$, then $(f^{-1})'(b) = \frac{1}{f'(a)}$.

General Radical to Fractional Exponent Rule

$\sqrt[n]{x^m} = x^{\frac{m}{n}}$, with examples including $\sqrt{x} = x^{\frac{1}{2}}$, $\sqrt[3]{x} = x^{\frac{1}{3}}$, and $\frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}$.

Unit 3 Study Priorities

The prioritized topics for review: 1. Chain Rule, 2. Product Rule, 3. Quotient Rule, 4. Implicit Differentiation, and 5. Inverse Functions.