Derivatives of Inverse Trig Functions — Definition, Formula & Examples

Derivatives of inverse trig functions are a set of six standard differentiation formulas that give the rate of change of arcsin, arccos, arctan, arcsec, arccsc, and arccot. Each derivative produces an algebraic expression (no trig functions remain), typically involving square roots or quadratic terms.

For each inverse trigonometric function $y = f^{-1}(x)$ defined on its principal branch, the derivative $\frac{dy}{dx}$ is obtained via implicit differentiation of the relation $x = f(y)$ , yielding formulas such as $\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}$ for $|x|<1$ . When the argument is a composite function $u(x)$ , the chain rule applies: multiply by $\frac{du}{dx}$ .

Key Formula

\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}[\arccos x] = \frac{-1}{\sqrt{1-x^2}}$$ $$\frac{d}{dx}[\arctan x] = \frac{1}{1+x^2}, \quad \frac{d}{dx}[\text{arccot}\, x] = \frac{-1}{1+x^2}$$ $$\frac{d}{dx}[\text{arcsec}\, x] = \frac{1}{|x|\sqrt{x^2-1}}, \quad \frac{d}{dx}[\text{arccsc}\, x] = \frac{-1}{|x|\sqrt{x^2-1}}

Where:

$x$ = The independent variable, restricted to the domain of each inverse trig function

Worked Example

Problem: Find the derivative of

y = \arctan(3x)

Identify the outer and inner functions: The outer function is arctan(u) and the inner function is u = 3x.

\frac{d}{dx}[\arctan u] = \frac{1}{1+u^2} \cdot \frac{du}{dx}

Apply the chain rule: Substitute u = 3x and compute du/dx = 3.

\frac{dy}{dx} = \frac{1}{1+(3x)^2} \cdot 3 = \frac{3}{1+9x^2}

Answer:

\dfrac{dy}{dx} = \dfrac{3}{1+9x^2}

Why It Matters

These formulas appear repeatedly on the AP Calculus AB/BC exams, both in differentiation problems and as recognizable forms in integration (since each derivative formula corresponds to an antiderivative rule). Engineers and physicists also encounter them when modeling angles in navigation, optics, and control systems.

Common Mistakes

Mistake: Forgetting the chain rule when differentiating expressions like arcsin(5x) and writing just 1/√(1−25x²).

Correction: You must multiply by the derivative of the inner function. The correct result is 5/√(1−25x²). Always ask: is the argument something other than plain x?