Unit 3 Formula Sheet: Limit Definitions and Derivative Rules

Vocabulary review flashcards covering limit definitions of derivatives and essential differentiation rules for algebraic, exponential, logarithmic, trigonometric, and inverse trigonometric functions.

Limit definition for the derivative at a point

$$f^{\prime}(a)=\frac{f(x) - f(a)}{x - a}$$

Limit definition for the general derivative

$$f^{\prime}(x)=\frac{f(x + h) - f(x)}{h}$$

Derivative of $x^n$

$$nx^{n-1}$$

Derivative of $e^x$

$$e^{x}$$

Derivative of $\sin(x)$

$$\cos(x)$$

Derivative of $\ln(x)$

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

Derivative of $\cos(x)$

$$-\sin(x)$$

Derivative of $a^x$

$$\ln(a)\cdot a^{x}$$

Derivative of $\tan(x)$

$$\sec^2(x)$$

Derivative of $\log_b(x)$

$$\frac{1}{x \cdot\ln(b)}$$

Derivative of $\sec(x)$

$$\sec(x)\tan(x)$$

Derivative of $\arcsin(x)$

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

Derivative of $\csc(x)$

$$-\csc(x)\cot(x)$$

Derivative of $\cos^{-1}(x)$

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

Derivative of $\cot(x)$

$$-\csc^2(x)$$

Derivative of $\tan^{-1}(x)$

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