Mathematics and Statistics · Ch 9 — Differentiation

Derivatives of Standard Functions

Differentiating every function from first principles would be slow, so the results for common functions are established once and memorised as a table of standard derivatives. Each can be proved from the definition, but here they are collected for use.

Algebraic:

\frac{d}{dx}(x^{n}) = n\,x^{n-1} \quad (\text{power rule, any real } n), \qquad \frac{d}{dx}(c) = 0 \quad (c \text{ constant}).

The power rule also handles roots and reciprocals: $\sqrt{x}=x^{1/2}$ so $\dfrac{d}{dx}\sqrt{x}=\dfrac{1}{2}x^{-1/2}=\dfrac{1}{2\sqrt{x}}$ , and $\dfrac1x = x^{-1}$ so $\dfrac{d}{dx}\dfrac1x = -x^{-2} = -\dfrac{1}{x^2}$ .

Exponential and logarithmic:

\frac{d}{dx}(e^{x}) = e^{x}, \qquad \frac{d}{dx}(a^{x}) = a^{x}\log_e a \ \ (a>0), \qquad \frac{d}{dx}(\log_e x) = \frac{1}{x}.

The exponential $e^x$ is the one function equal to its own derivative — a fact used constantly.

Trigonometric (angle in radians):

\frac{d}{dx}(\sin x) = \cos x, \qquad \frac{d}{dx}(\cos x) = -\sin x, \qquad \frac{d}{dx}(\tan x) = \sec^{2} x.

Note

The Power Rule Covers Roots and Reciprocals Too …

Definition 6Power rule

$\frac{d}{dx}(x^n)=n\,x^{n-1}$ for any real $n$ . Covers roots ( $x^{1/2}$ ) and reciprocals ( $x^{-1}$ ) once they ar …

Definition 7Derivative of a constant

$\frac{d}{dx}(c)=0$ : a constant does not change, so its rate of change is zero (a horizontal lin …

Definition 8Exponential/logarithmic derivatives

$\frac{d}{dx}(e^x)=e^x$ , $\frac{d}{dx}(a^x)=a^x\log_e a$ , $\frac{d}{dx}(\log_ …

Definition 9Trigonometric derivatives (radians)

$\frac{d}{dx}(\sin x)=\cos x$ , $\frac{d}{dx}(\cos x)=-\sin x$ , $\frac{d}{dx}(\tan x)=\sec^2 x$ . Valid only whe …

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