Derivative Rules

\frac{d}{dx} (x^n)

nx^{n-1}

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

e^x

\frac{d}{dx} (f + g)

f' + g'

\frac{d}{dx} (f - g)

f' - g'

\frac{d}{dx} (fg)

fg' + gf'

\frac{d}{dx} \left(\frac{f}{g}\right)

\frac{g f' - f g'}{g^2}

\frac{d}{dx}(f(g(x))

f'(g(x))g'(x)

\frac{d}{dx} (cx^n)

cnx^{n-1}

\frac{d}{dx} (\sin x)

\cos x

\frac{d}{dx} (\cos x)

-\sin x

\frac{d}{dx} (\tan x)

\sec^2(x)

\frac{d}{dx} (\sec x)

\sec x \tan x

\frac{d}{dx} (\csc x)

-\csc x \cot x

\frac{d}{dx} (\cot x)

-\csc^2(x)

\frac{d}{dx} (a^x)

a^x \ln a

Definition of f'(x)

\lim_{h\to 0} \frac{f(x+h) - f(x)}{h}

\frac{d}{dx} (\sin(2x)) - Double Angle

2\sin x \cos x

\frac{d}{dx} (c)

0

\frac{d}{dx} (\ln |x|)

\frac{1}{x}

\frac{d}{dx} \log_a(x)

\frac{1}{x \ln a}

\frac{d}{dx} \text{ (position function)}

velocity

\frac{d}{dx} \text{ (acceleration function)}

jerk

\frac{d}{dx} \text{ (velocity function)}

acceleration

\lim_{x\to 0} \frac{\sin x}{x}

1

\frac{d}{dx} (\sin^{-1}(x))

\frac{1}{\sqrt{1-x^2}}

\frac{d}{dx} (\cos^{-1}(x))

-\frac{1}{\sqrt{1-x^2}}

\frac{d}{dx} (\tan^{-1}(x))

\frac{1}{1+x^2}

\frac{d}{dx} (\sec^{-1}(x))

\frac{1}{|x|\sqrt{x^2-1}}

\frac{d}{dx} (\csc^{-1}(x))

-\frac{1}{|x|\sqrt{x^2-1}}

\frac{d}{dx} (\cot^{-1}(x))

-\frac{1}{1+x^2}

average velocity

\frac{\Delta y}{\Delta x}

Instantaneous velocity

\lim_{\Delta x\to 0} \frac{\Delta y}{\Delta x}