Derivative of cot x: Formula, Proof and Examples
The derivative of cot x is -csc²x. See the quotient rule proof, the -(1 + cot²x) form, the chain rule for cot(u) and examples like cot(2x).
Trigonometric derivatives Updated 2 min read
At a glance
- Derivative
- -\csc^2 x for x \ne k\pi
- Other forms
- -\dfrac{1}{\sin^2 x} = -(1 + \cot^2 x)
- Chain rule form
- \dfrac{d}{dx}\cot u = -\csc^2 u \cdot u'
- Slope at x = π/2
- -\csc^2\frac{\pi}{2} = -1
- Integral version
- \int \csc^2 x\,dx = -\cot x + C
On this page
The derivative of \cot x is -\csc^2 x. It is the "co" partner of \frac{d}{dx}\tan x = \sec^2 x: swap to the co-functions and add a minus sign.
Graph of cot x and its derivative
Proof with the quotient rule
Write \cot x = \frac{\cos x}{\sin x} and use the quotient rule:
The top is -1, so
A second proof from tan x
Since \cot x = (\tan x)^{-1}, the chain rule and the derivative of tan x give
Same result. The identity \csc^2 x = 1 + \cot^2 x also lets you write the answer as -(1 + \cot^2 x).
The chain rule version: derivative of cot(u)
| Function | Inside u | Derivative |
|---|---|---|
| \cot(2x) | 2x | -2\csc^2(2x) |
| \cot(x^3) | x^3 | -3x^2\csc^2(x^3) |
| \cot\sqrt{x} | \sqrt{x} | -\frac{\csc^2\sqrt{x}}{2\sqrt{x}} |
| \cot(\pi - x) | \pi - x | \csc^2(\pi - x) |
Worked examples
Example 1: a power. \frac{d}{dx}\cot^2 x = 2\cot x\cdot(-\csc^2 x) = -2\cot x\csc^2 x.
Example 2: a product. \frac{d}{dx}(x\cot x) = \cot x - x\csc^2 x.
Example 3: a value. At x = \frac{\pi}{4}, \csc\frac{\pi}{4} = \sqrt{2}, so the slope is -(\sqrt{2})^2 = -2. The tangent line through \left(\frac{\pi}{4}, 1\right) is y = 1 - 2\left(x - \frac{\pi}{4}\right).
Example 4: the log of sine. \frac{d}{dx}\ln(\sin x) = \frac{\cos x}{\sin x} = \cot x. Differentiating again gives -\csc^2 x. This is why \int\cot x\,dx = \ln\lvert\sin x\rvert + C, see the integral of cot x.
How to remember it
Pair \cot x with \tan x. The derivative of tangent is \sec^2 x. Swap each function for its "co" partner, tangent to cotangent and secant to cosecant, then put a minus sign in front. That gives -\csc^2 x with no proof needed. The same swap works for every pair of trig derivatives, so you only need to memorize three of the six results. A quick graph check confirms the sign: each branch of cotangent falls from left to right, so its slope must always be negative. The steepness grows near the asymptotes at multiples of \pi, where \csc^2 x blows up.
Common mistakes
- Dropping the minus sign. Cotangent always decreases, so its derivative must be negative.
- Writing -\csc x\cot x. That is the derivative of csc x, a different result.
- Writing -\sec^2 x. The square goes on cosecant, the partner of secant.
For the whole family in one place, read derivatives of trig functions.
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