Derivative Calculator

Inverse trigonometric rule

The inverse trigonometric functions differentiate to algebraic expressions — no trigonometry left in the answer.

\[\frac{d}{dx}\arctan x = \frac{1}{1 + x^{2}}\]

What it says

Each one comes from the inverse function rule. For \(y = \arcsin x\) we have \(\sin y = x\); differentiating both sides gives \(\cos y \cdot y' = 1\), so \(y' = \frac{1}{\cos y} = \frac{1}{\sqrt{1 - x^{2}}}\), using \(\cos y = \sqrt{1 - \sin^{2}y}\).

The table: \(\left(\arcsin x\right)' = \frac{1}{\sqrt{1 - x^{2}}}\), \(\left(\arccos x\right)' = -\frac{1}{\sqrt{1 - x^{2}}}\), \(\left(\arctan x\right)' = \frac{1}{1 + x^{2}}\). The arc-co-functions are the negatives of their partners, because \(\arcsin x + \arccos x = \frac{\pi}{2}\) is constant.

That these derivatives are purely algebraic is why inverse trigonometric functions turn up as the answers to integrals that look nothing like trigonometry.

When it applies

  • \(\arcsin\), \(\arccos\), \(\arctan\) and their reciprocal partners.
  • Written either way in this calculator: \(\texttt{arctan(x)}\) or \(\texttt{tan\^-1(x)}\).
  • With the chain rule for a compound argument, as in \(\arctan(3x)\).

Five worked examples

Every line is the step the calculator would show, in the order it applies them. Each graph is live: hover it to read both curves and see the tangent whose slope is the derivative, drag to pan, scroll to zoom.

Example 1
\[\frac{d}{dx}\left[\arcsin\left(x\right)\right]\]

  1. From \(\sin y = x\), differentiate both sides.

    \[\cos y \cdot y' = 1\]

  2. Substitute \(\cos y = \sqrt{1 - x^{2}}\).

    \[y' = \frac{1}{\sqrt{1 - x^{2}}}\]
Answer
\[\frac{1}{\sqrt{1 - x^{2}}}\]

Both curves exist only on \([-1, 1]\), which is why the graph frames itself so tightly. f′ is a positive well that blows up at the two ends, where arcsine turns vertical, and reaches its lowest value of 1 at the origin.

Defined for \(-1 < x < 1\), and blowing up at both ends where the sine curve turns over.

Open arcsin(x) in the calculator
Example 2
\[\frac{d}{dx}\left[\arccos\left(x\right)\right]\]

  1. Same derivation, or note that \(\arccos x = \frac{\pi}{2} - \arcsin x\).

    \[= -\frac{1}{\sqrt{1 - x^{2}}}\]
Answer
\[-\frac{1}{\sqrt{1 - x^{2}}}\]

The mirror image: f falls from \(\pi\) to 0 while f′ is the same well flipped below the axis.

Exactly the negative of the arcsine derivative, since the two functions add to a constant.

Open arccos(x) in the calculator
Example 3
\[\frac{d}{dx}\left[\arctan\left(x\right)\right]\]

  1. From \(\tan y = x\).

    \[\sec^{2}y \cdot y' = 1\]

  2. And \(\sec^{2}y = 1 + \tan^{2}y = 1 + x^{2}\).

    \[y' = \frac{1}{1 + x^{2}}\]
Answer
\[\frac{1}{1 + x^{2}}\]

f levels off toward \(\pm\tfrac{\pi}{2}\) at both ends, and f′ is a bump of height 1 over the origin. Nothing blows up anywhere.

Defined for every real \(x\), which is why \(\arctan\) is such a well-behaved function.

Open arctan(x) in the calculator
Example 4
\[\frac{d}{dx}\left[\arctan\left(3 x\right)\right]\]

  1. Chain rule with \(u = 3x\).

    \[= \frac{1}{1 + \left(3x\right)^{2}}\cdot 3\]

  2. Square the inside.

    \[= \frac{3}{1 + 9x^{2}}\]
Answer
\[\frac{3}{1 + 9 x^{2}}\]

The same S-curve squeezed threefold, so f′ is three times taller and three times narrower.

The \(3\) is squared in the denominator but not in the numerator — a common place to slip.

Open arctan(3x) in the calculator
Example 5
\[\frac{d}{dx}\left[x \arctan\left(x\right)\right]\]

  1. Product rule.

    \[= 1 \cdot \arctan x + x\cdot\frac{1}{1 + x^{2}}\]

  2. Tidy.

    \[= \arctan x + \frac{x}{1 + x^{2}}\]
Answer
\[\arctan\left(x\right) + \frac{x}{1 + x^{2}}\]

A smooth valley with its floor at the origin, where f′ changes sign. Far out, f grows almost linearly.

The pairing behind \(\int \arctan x\,dx = x\arctan x - \tfrac{1}{2}\ln(1 + x^{2})\).

Open x arctan(x) in the calculator

Watch out for

  • \(\tan^{-1}x\) means \(\arctan x\), not \(\frac{1}{\tan x}\).
  • The minus sign belongs to \(\arccos\) and \(\text{arccot}\), not to \(\arcsin\) or \(\arctan\).
  • \(\arcsin\) and \(\arccos\) are only defined on \([-1, 1]\), and their derivatives only inside it.
Trigonometric ruleChain ruleInverse hyperbolic rule