Hard

Derivatives of Tangent and Other Trigonometric Functions

Simple Explanation

The remaining four trigonometric functions have their own derivative formulas: d/dx[tanx]=sec²x, d/dx[cotx]=−csc²x, d/dx[secx]=secx·tanx, and d/dx[cscx]=−cscx·cotx.

Why Do We Need It?

These complete the full set of trigonometric derivatives, all of which can actually be derived from the sine and cosine derivatives using the Quotient Rule.

Formula

Derivatives of Tangent, Cotangent, Secant, and Cosecant

d/dx[tan x]=sec²x, d/dx[cot x]=−csc²x, d/dx[sec x]=sec x·tan x, d/dx[csc x]=−csc x·cot x

The derivatives of the remaining four trigonometric functions, each expressible using the reciprocal trig functions.

x: — the input angle, in radians

When to use it: Whenever differentiating an expression involving tangent, cotangent, secant, or cosecant.

Worked Example

Evaluate the derivative of tangent at a point

Find the derivative of f(x)=tanx at x=0.

Why Does This Work?

tan x = sinx/cosx, so applying the Quotient Rule gives (cosx·cosx − sinx·(−sinx))/cos²x = (cos²x+sin²x)/cos²x = 1/cos²x (using the Pythagorean identity) = sec²x — exactly the stated derivative, derived directly from the sine and cosine derivatives.

Real-Life Example

Analyzing the rate of change of a shadow's length

The length of a shadow, related to the tangent of the sun's elevation angle, changes at a certain rate as the sun moves across the sky.

The derivative of tangent gives exactly this rate of change with respect to the changing angle.

Practice

Find the derivative of f(x)=sec(x) at x=0.

Hard

Common mistake

Forgetting the negative signs in the derivatives of cotangent and cosecant — only tangent and secant have positive derivative formulas among these four.

Quick Review

d/dx[tanx]=sec²x and d/dx[secx]=secx·tanx (both positive).
d/dx[cotx]=−csc²x and d/dx[cscx]=−cscx·cotx (both negative).
All four can be derived from the sine and cosine derivatives via the Quotient Rule.