Power, linearity, and elementary derivative rules

Understand the idea

Use the power rule on powers, including negative and fractional powers where defined. Differentiate constants to zero and distribute differentiation through finite sums. Memorize derivatives of the six trigonometric functions and the natural exponential and logarithm. Trigonometric formulas assume radians; domain restrictions remain in force after differentiating.

\frac d{dx}x^n=nx^{n-1},\quad (e^x)'=e^x,\quad(\ln x)'=\frac1x,\quad(\sin x)'=\cos x,\quad(\cos x)'=-\sin x,\quad(\tan x)'=\sec^2x

Worked example

Find the derivative of $\displaystyle f(x)=3x^4-2/x+5\sin x$ .

Rewrite

\displaystyle -2/x=-2x^{-1}

. Differentiate term by term to obtain

\displaystyle f'(x)=12x^3+2x^{-2}+5\cos x

, on the original domain

\displaystyle x\ne0

.

Try it yourself

Differentiate $\displaystyle g(x)=\sec x+\cot x+\csc x$ .

Show solution and reasoning

$\displaystyle g'(x)=\sec x\tan x-\csc^2x-\csc x\cot x$ wherever the original terms are defined.