Calculus Differentiation Rules and Derivatives of Common Functions

A complete collection of vocabulary flashcards for fundamental differentiation rules, composite functions, and elementary function derivatives along with their respective domains.

Derivative of a Sum $u(x) + v(x)$

$$u'(x) + v'(x)$$

Derivative of Constant Multiple $k u(x)$

$k u'(x)$ for $k \in \mathbb{R}$

Derivative of a Product $u(x) v(x)$

$$u'(x) v(x) + u(x) v'(x)$$

Derivative of Reciprocal Function $\frac{1}{v(x)}$

$-\frac{v'(x)}{(v(x))^2}$ given $v(x) \neq 0$

Derivative of a Quotient $\frac{u(x)}{v(x)}$

$\frac{u'(x) v(x) - u(x) v'(x)}{(v(x))^2}$ given $v(x) \neq 0$

Derivative of Composite Power $u^n(x) = (u(x))^n$

$n (u(x))^{n-1} u'(x)$ for $n \in \mathbb{N}$

Derivative of Composite Square Root $\sqrt{u(x)}$

$$\frac{u'(x)}{2 \sqrt{u(x)}}$$

Derivative of Composite Exponential $e^{u(x)}$

$$u'(x) e^{u(x)}$$

Derivative of Composite Natural Logarithm $\ln(u(x))$

$$\frac{u'(x)}{u(x)}$$

Derivative of Composite Sine $\sin(u(x))$

$$u'(x) \cos(u(x))$$

Derivative of Composite Cosine $\cos(u(x))$

$$-u'(x) \sin(u(x))$$

Linear Function $f(x) = a x + b$

Domain $\mathcal{D}_f = \mathbb{R}$ ($a, b \in \mathbb{R}$), derivative $f'(x) = a$ with derivative domain $\mathcal{D}_{f'} = \mathbb{R}$

Power Function $f(x) = x^n$

Domain $\mathcal{D}_f = \mathbb{R}$ ($n \in \mathbb{N}$), derivative $f'(x) = n x^{n-1}$ with derivative domain $\mathcal{D}_{f'} = \mathbb{R}$

Reciprocal Power Function $f(x) = \frac{1}{x^n} = x^{-n}$

Domain $\mathcal{D}_f = \mathbb{R}^*$ ($n \in \mathbb{N}^*$), derivative $f'(x) = -\frac{n}{x^{n+1}}$ with derivative domain $\mathcal{D}_{f'} = \mathbb{R}^*$

Square Root Function $f(x) = \sqrt{x} = x^{1/2}$

Domain $\mathcal{D}_f = \mathbb{R}^+$, derivative $f'(x) = \frac{1}{2 \sqrt{x}}$ with derivative domain $\mathcal{D}_{f'} = \mathbb{R}^{*+}$

Exponential Function $f(x) = e^x$

Domain $\mathcal{D}_f = \mathbb{R}$, derivative $f'(x) = e^x$ with derivative domain $\mathcal{D}_{f'} = \mathbb{R}$

Absolute Natural Logarithm Function $f(x) = \ln(|x|)$

Domain $\mathcal{D}_f = \mathbb{R}^*$, derivative $f'(x) = \frac{1}{x}$ with derivative domain $\mathcal{D}_{f'} = \mathbb{R}^*$

Sine Function $f(x) = \sin(x)$

Domain $\mathcal{D}_f = \mathbb{R}$, derivative $f'(x) = \cos(x)$ with derivative domain $\mathcal{D}_{f'} = \mathbb{R}$

Cosine Function $f(x) = \cos(x)$

Domain $\mathcal{D}_f = \mathbb{R}$, derivative $f'(x) = -\sin(x)$ with derivative domain $\mathcal{D}_{f'} = \mathbb{R}$