Calculus: Differentiation Rules for Trigonometric, Inverse Trigonometric, Exponential, and Logarithmic Functions

Comprehensive vocabulary flashcards covering the differentiation rules and Chain Rule formulas for basic trigonometric functions, inverse trigonometric functions, exponential functions, and logarithmic functions.

Chain Rule

$$(f \text{ o } g)'(x) = f'(g(x)) \times g'(x)$$

Derivative of $\sin(x)$

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

Derivative of $\cos(x)$

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

Derivative of $\tan(x)$

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

Derivative of $\cot(x)$

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

Derivative of $\sec(x)$

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

Derivative of $\csc(x)$

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

Derivative of $\sin^{-1}(u)$

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

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

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

Derivative of $\sec^{-1}(u)$

$$\frac{d}{dx}\,\sec^{-1}(u)=\frac{u'}{|u|\sqrt{u^2-1}}$$

Derivative of $e^u$

$$\frac{d}{dx}\,e^u = e^u \times \frac{du}{dx}$$

Derivative of $a^u$ ($a > 0$, $a \neq 1$)

$$\frac{d}{dx}\,a^u = a^u\,\ln(a) \times \frac{du}{dx}$$

Derivative of $\ln(u)$

$$\frac{d}{dx}\,\ln(u)=\frac{u'}{u}$$

Derivative of $\log_a(u)$

$$\frac{d}{dx}\,\log_{a}(u)=\frac{u'}{u\,\ln(a)}$$