Derivatives of Trigonometric and Inverse Functions

In calculus, memorizing the derivatives of standard functions is essential. This guide covers the derivatives of trigonometric functions, inverse trigonometric functions, and general exponential and logarithmic functions.

Derivatives of Trigonometric Functions

Here are the derivatives of the six basic trigonometric functions:

• $\frac{d}{dx}(\sin x) = \cos x$
• $\frac{d}{dx}(\cos x) = -\sin x$
• $\frac{d}{dx}(\tan x) = \sec^2 x$
• $\frac{d}{dx}(\cot x) = -\csc^2 x$
• $\frac{d}{dx}(\sec x) = \sec x \tan x$
• $\frac{d}{dx}(\csc x) = -\csc x \cot x$

Derivatives of Inverse Trigonometric Functions

The inverse trigonometric functions (often written as $\arcsin x$ or $\sin^{-1} x$) have specific derivative rules:

• $\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 - x^2}}$
• $\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1 - x^2}}$
• $\frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2}$

(Note: $\text{arccot}$, $\text{arcsec}$, and $\text{arccsc}$ follow similar patterns, but these three are the most commonly tested).

General Exponential and Logarithmic Functions

For exponential and logarithmic functions with bases other than $e$, we adjust the derivative by multiplying or dividing by the natural logarithm of the base ($\ln a$):

• Exponential Function: $\frac{d}{dx}(a^x) = a^x \ln a$
• Logarithmic Function: $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$

Applying the Chain Rule

When the input is a function $u(x)$ rather than just $x$, you must apply the Chain Rule: multiply the outside derivative by the derivative of the inside function, $u'(x)$.

Example 1: Find $\frac{d}{dx}\tan(3x)$

The outer function is $\tan(u)$ and the inner function is $u = 3x$. $\frac{d}{dx}\tan(3x) = \sec^2(3x) \cdot \frac{d}{dx}(3x) = 3\sec^2(3x)$

Example 2: Find $\frac{d}{dx}\arcsin(2x)$

The outer function is $\arcsin(u)$ and the inner function is $u = 2x$. $\frac{d}{dx}\arcsin(2x) = \frac{1}{\sqrt{1 - (2x)^2}} \cdot \frac{d}{dx}(2x) = \frac{2}{\sqrt{1 - 4x^2}}$