Exponential, Logarithmic and Trigonometric Derivatives

Learn the structure behind a formula table

The exponential $e^x$ is distinguished by having its own value as its rate of change. Other positive constant bases introduce a scale factor: $(a^x)'=a^x\ln a$ . The logarithm reverses the exponential, and inverse differentiation gives $(\ln x)'=1/x$ for $x>0$ .

The usual trigonometric derivatives assume radians. This is tied to $\sin h/h\to1$ as $h\to0$ . Differentiating a degree-based function requires the extra conversion factor $\pi/180$ .

Visual guide

VISUAL GUIDEExponential growth and logarithmic slope

Exponential

eˣ
Tangent y = 1 + x

Logarithm

ln x
Tangent y = x − 1

The tangent to eˣ at zero has slope 1. The tangent to ln x at one also has slope 1, but the functions change differently away from those points. The logarithm exists only for positive inputs.

The working formulas

Function	Derivative	Real-domain restriction
$e^x$	$e^x$	all real $x$
$a^x$	$a^x\ln a$	fixed $a>0$
$\ln\lvert x\rvert$	$1/x$	$x\ne0$
$\sin x$	$\cos x$	radians
$\cos x$	$-\sin x$	radians
$\tan x$	$\sec^2x$	$\cos x\ne0$
$\cot x$	$-\csc^2x$	$\sin x\ne0$
$\sec x$	$\sec x\tan x$	$\cos x\ne0$
$\csc x$	$-\csc x\cot x$	$\sin x\ne0$
$\arcsin x$	$1/\sqrt{1-x^2}$	$\lvert x\rvert<1$
$\arccos x$	$-1/\sqrt{1-x^2}$	$\lvert x\rvert<1$
$\arctan x$	$1/(1+x^2)$	all real $x$

These are outer-function derivatives. A nontrivial input still requires the chain rule. For instance, $\ln|u|$ differentiates to $u'/u$ on any interval where $u$ is differentiable and nonzero.

Worked example: a decaying oscillation

For $y=e^{-2t}\sin(3t)$ , combine product and chain rules:

y'=e^{-2t}\bigl(3\cos(3t)-2\sin(3t)\bigr).

At $t=0$ , the value is zero and the rate is $3$ . The exponential envelope is decreasing, but that does not force the whole function to decrease at every instant. The oscillation's contribution matters too.

Worked example: an inverse-trig composition

For $f(x)=\arctan(2x)$ , the outer derivative is $1/[1+(2x)^2]$ and the inner derivative is $2$ . Thus $f'(x)=2/(1+4x^2)$ , always positive. For $g(x)=\ln(x^2-4)$ , the derivative is $2x/(x^2-4)$ but the original real function exists only for $|x|>2$ . The derivative's formula alone would not reveal that restriction.

Practice

Differentiate $5^{2x}$ .
Differentiate $\cos(x^3)$ .
Differentiate $\arcsin(2x)$ and state where the derivative is finite.

Show worked solutions

$2\ln(5)\,5^{2x}$ , using both the base factor and chain factor.
$-3x^2\sin(x^3)$ .
$2/\sqrt{1-4x^2}$ for $|x|<1/2$ . The function also has endpoint values, but no finite derivative at those endpoints.