What Are the Derivatives of Tan x, Sec x, Arcsin x, Arccos x, and Arctan x?

The required derivatives are listed below. These are IB Mathematics: Analysis and Approaches HL results from AHL 5.15: Further derivatives and indefinite integration, and they apply when angles are measured in radians.

These formulas are not isolated facts. Each describes the gradient of a trigonometric or inverse-trigonometric function, and each becomes a template for differentiating a composite function. The square-root denominators also explain why the derivatives of arcsine and arccos are undefined at the endpoints $x=\pm 1$, even though both original functions are defined there.

Reasoning and Mechanism

For tangent, write $\tan x=\dfrac{\sin x}{\cos x}$. Applying the quotient rule gives a numerator of $\cos^2x+\sin^2x=1$ and a denominator of , producing . For secant, rewrite ; the chain rule then simplifies the derivative to .

The inverse-trigonometric results follow from implicit differentiation. If $y=\arcsin x$, then $x=\sin y$, so

$$1=\cos y\dfrac{dy}{dx}.$$

Since $\cos y=\sqrt{1-x^2}$ on the principal range of arcsine, the derivative follows. Starting with $$ explains the negative sign for arccos. Starting with and using gives the arctangent result.

For composite functions, apply the chain rule and multiply by the inner derivative:

$$\dfrac{d}{dx}\tan(3x)=3\sec^2(3x),\qquad \dfrac{d}{dx}\arcsin(2x)=\dfrac{2}{\sqrt{1-4x^2}}.$$

The arcsine composite's derivative exists only for $|x|<\dfrac{1}{2}$. More generally, replace $x$ by $g(x)$ in the basic formula and multiply by (g'(x)).

For example, let $f(x)=\sec(2x^2)$. The outer derivative is secant times tangent, while the inner derivative is $4x$. Therefore,

$$f'(x)=4x\sec(2x^2)\tan(2x^2).$$

Writing the inner derivative explicitly prevents the most common chain-rule omission.

Before differentiating an inverse-trigonometric composite, check its domain. For $\arcsin(g(x))$, the original function requires $-1\leq g(x)\leq 1$, while its derivative requires strict inequalities. This distinction is often tested through domain questions.

Exam Technique

A common misconception is that $\arcsin x$ means $1/\sin x$. It does not: arcsine is an inverse function, whereas the reciprocal of sine is cosecant. In AA HL Paper 1 or Paper 2, show the chain-rule factor, retain brackets around composite arguments, use radians, and never omit the negative sign in the derivative of arccos.