Derivative Rules for Elementary, Trigonometric, and Hyperbolic Functions

Comprehensive practice flashcards covering derivative rules for exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions.

What is the derivative of $e^x$ with respect to $x$?

$$\frac{d}{dx}(e^x) = e^x$$

What is the derivative of $a^x$ with respect to $x$?

$$\frac{d}{dx}(a^x) = a^x \ln(a)$$

Using the chain rule, what is the derivative of $e^{g(x)}$ with respect to $x$?

$$\frac{d}{dx}(e^{g(x)}) = e^{g(x)} g'(x)$$

Using the chain rule, what is the derivative of $a^{g(x)}$ with respect to $x$?

$$\frac{d}{dx}(a^{g(x)}) = \ln(a) a^{g(x)} g'(x)$$

What is the derivative of $\ln(x)$ for $x > 0$?

$$\frac{d}{dx}(\ln(x)) = \frac{1}{x}$$

Using the chain rule, what is the derivative of $\ln(g(x))$ with respect to $x$?

$$\frac{d}{dx}(\ln(g(x))) = \frac{g'(x)}{g(x)}$$

What is the derivative of $\log_a(x)$ for $x > 0$?

$$\frac{d}{dx}(\log_a(x)) = \frac{1}{x \ln(a)}$$

Using the chain rule, what is the derivative of $\log_a(g(x))$ with respect to $x$?

$$\frac{d}{dx}(\log_a(g(x))) = \frac{g'(x)}{g(x) \ln(a)}$$

What is the derivative of $\sin(x)$ with respect to $x$?

$$\frac{d}{dx}(\sin(x)) = \cos(x)$$

What is the derivative of $\cos(x)$ with respect to $x$?

$$\frac{d}{dx}(\cos(x)) = -\sin(x)$$

What is the derivative of $\tan(x)$ with respect to $x$?

$$\frac{d}{dx}(\tan(x)) = \sec^2(x)$$

What is the derivative of $\cot(x)$ with respect to $x$?

$$\frac{d}{dx}(\cot(x)) = -\csc^2(x)$$

What is the derivative of $\sec(x)$ with respect to $x$?

$\frac{d}{dx}(\sec(x)) = \sec(x)\tan(x)$ (or $\sec^2(x)\sin(x)$)

What is the derivative of $\csc(x)$ with respect to $x$?

$\frac{d}{dx}(\csc(x)) = -\csc(x)\cot(x)$ (or $-\csc^2(x)\cos(x)$)

What is the derivative of $\sin^{-1}(x)$ for $x \neq \pm 1$?

$$\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$$

What is the derivative of $\cos^{-1}(x)$ for $x \neq \pm 1$?

$$\frac{d}{dx}(\cos^{-1}(x)) = \frac{-1}{\sqrt{1-x^2}}$$

What is the derivative of $\tan^{-1}(x)$ with respect to $x$?

$$\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2}$$

What is the derivative of $\cot^{-1}(x)$ with respect to $x$?

$$\frac{d}{dx}(\cot^{-1}(x)) = \frac{-1}{1+x^2}$$

What is the derivative of $\sec^{-1}(x)$ for $x \neq \pm 1, 0$?

$$\frac{d}{dx}(\sec^{-1}(x)) = \frac{1}{x\sqrt{x^2-1}}$$

What is the derivative of $\csc^{-1}(x)$ for $x \neq \pm 1, 0$?

$$\frac{d}{dx}(\csc^{-1}(x)) = \frac{-1}{x\sqrt{x^2-1}}$$

What is the derivative of $\sinh(x)$ with respect to $x$?

$$\frac{d}{dx}(\sinh(x)) = \cosh(x)$$

What is the derivative of $\cosh(x)$ with respect to $x$?

$$\frac{d}{dx}(\cosh(x)) = \sinh(x)$$

What is the derivative of $\tanh(x)$ with respect to $x$?

$$\frac{d}{dx}(\tanh(x)) = \text{sech}^2(x)$$

What is the derivative of $\text{csch}(x)$ with respect to $x$?

$$\frac{d}{dx}(\text{csch}(x)) = -\text{csch}(x)\coth(x)$$

What is the derivative of $\text{sech}(x)$ with respect to $x$?

$$\frac{d}{dx}(\text{sech}(x)) = -\text{sech}(x)\tanh(x)$$

What is the derivative of $\coth(x)$ with respect to $x$?

$$\frac{d}{dx}(\coth(x)) = -\text{csch}^2(x)$$

What is the derivative of $\sinh^{-1}(x)$ with respect to $x$?

$$\frac{d}{dx}(\sinh^{-1}(x)) = \frac{1}{\sqrt{1+x^2}}$$

What is the derivative of $\cosh^{-1}(x)$ for $x > 1$?

$$\frac{d}{dx}(\cosh^{-1}(x)) = \frac{1}{\sqrt{x^2-1}}$$

What is the derivative of $\tanh^{-1}(x)$ for $|x| < 1$?

$$\frac{d}{dx}(\tanh^{-1}(x)) = \frac{1}{1-x^2}$$

What is the derivative of $\text{csch}^{-1}(x)$ for $x \neq 0$?

$$\frac{d}{dx}(\text{csch}^{-1}(x)) = \frac{-1}{|x|\sqrt{1+x^2}}$$

What is the derivative of $\text{sech}^{-1}(x)$ for $0 < x < 1$?

$$\frac{d}{dx}(\text{sech}^{-1}(x)) = \frac{-1}{x\sqrt{1-x^2}}$$

What is the derivative of $\coth^{-1}(x)$ for $|x| > 1$?

$$\frac{d}{dx}(\coth^{-1}(x)) = \frac{1}{1-x^2}$$