Basic Functions and Their Derivatives

Conditions for using the tables

The derivative reference sheet is a lookup aid; apply each formula only where the derivative exists.

• $\ln x$ and $\log_a x$ require $x>0$. For $a^x$, take $a>0$; for $\log_a x$, also require $a\ne1$.
• All trigonometric arguments are in radians. Tangent and secant exclude $\cos x=0$; cotangent and cosecant exclude $\sin x=0$.
• $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$ mean the principal inverse functions (arcsine, arccosine and arctangent), not reciprocals. Their ranges are respectively $[-\pi/2,\pi/2]$, $[0,\pi]$, and $(-\pi/2,\pi/2)$. Arcsine and arccosine are defined on $[-1,1]$, but their displayed derivatives require $-1<x<1$. Arctangent and its derivative are defined for all real inputs.
• In chain-rule variants, the inner function must be differentiable, and its output must lie where the outer function is differentiable. In particular, $\ln(f(x))$ requires $f(x)>0$, and a reciprocal requires $g(x)\ne0$.
• For a real constant $n$, the power formula is valid for positive bases. Integer powers allow the extensions described in the rule conditions; endpoints need separate checks.

A useful extension is $(\ln|f(x)|)'=f'(x)/f(x)$ wherever $f$ is differentiable and nonzero. For example, $\ln(x^2+1)$ has derivative $2x/(x^2+1)$ for every real $x$, whereas $\ln x$ cannot be evaluated at negative $x$ even though the expression $1/x$ can.

Standard Functions

Function Type	Function	Derivative
Constant Function	$f(x) = c$	$f'(x) = 0$
Linear Function	$f(x) = mx + b$	$f'(x) = m$
Quadratic Function	$f(x) = ax^2 + bx + c$	$f'(x) = 2ax + b$
Exponential Function (base e)	$f(x) = e^x$	$f'(x) = e^x$
Logarithmic Function (base e)	$f(x) = \ln(x)$	$f'(x) = \frac{1}{x}$
Exponential Function (general base)	$f(x) = a^x$	$f'(x) = a^x \ln(a)$
Logarithmic Function (general base)	$f(x) = \log_a(x)$	$f'(x) = \frac{1}{x \ln(a)}$
Derivative of a Square	$\frac{d}{dx} [f(x)]^2$	$2 f(x) f'(x)$
Derivative of a Reciprocal	$\frac{d}{dx} \left(\frac{1}{g(x)}\right)$	$-\frac{g'(x)}{[g(x)]^2}$

Trigonometric Functions

Function Type	Function	Derivative
Sine Function	$f(x) = \sin(x)$	$f'(x) = \cos(x)$
Cosine Function	$f(x) = \cos(x)$	$f'(x) = -\sin(x)$
Tangent Function	$f(x) = \tan(x)$	$f'(x) = \sec^2(x)$
Cotangent Function	$f(x) = \cot(x)$	$f'(x) = -\csc^2(x)$
Secant Function	$f(x) = \sec(x)$	$f'(x) = \sec(x)\tan(x)$
Cosecant Function	$f(x) = \csc(x)$	$f'(x) = -\csc(x)\cot(x)$
Inverse Sine Function	$f(x) = \sin^{-1}(x)$	$f'(x) = \frac{1}{\sqrt{1-x^2}}$
Inverse Cosine Function	$f(x) = \cos^{-1}(x)$	$f'(x) = -\frac{1}{\sqrt{1-x^2}}$
Inverse Tangent Function	$f(x) = \tan^{-1}(x)$	$f'(x) = \frac{1}{1+x^2}$

Chain Rule Variants

Chain Rule Derivative
$\frac{d}{dx} f(x)^n = n f(x)^{n-1} \cdot f'(x)$
$\frac{d}{dx} e^{f(x)} = e^{f(x)} \cdot f'(x)$
$\frac{d}{dx} \ln(f(x)) = \frac{f'(x)}{f(x)}$
$\frac{d}{dx} \sin(f(x)) = \cos(f(x)) \cdot f'(x)$
$\frac{d}{dx} \cos(f(x)) = -\sin(f(x)) \cdot f'(x)$
$\frac{d}{dx} \tan(f(x)) = \sec^2(f(x)) \cdot f'(x)$
$\frac{d}{dx} \sec(f(x)) = \sec(f(x)) \tan(f(x)) \cdot f'(x)$
$\frac{d}{dx} \tan^{-1}(f(x)) = \frac{f'(x)}{1 + f(x)^2}$

References

• tutorial.math.lamar.edu/pdf/calculus_cheat_sheet_derivatives.pdf
• personal.math.ubc.ca/~CLP/CLP1/clp_1_dc_text.pdf
• mitres_18_001_f17_full_book.pdf

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