# How do you find the derivative of cot x?

In a right-angled triangle, the cotangent of an angle is defined as:

The ratio of the length of the adjacent side to the length of the side opposite of the angle.

### Answer: Derivative of cot x is −Cosec^{2}x

We can proceed step by step to find the derivative.

**Explanation:**

As per trigonometric identities, cot x can also be written as cosx / sinx ------- (i)

Now we can use quotient rule of differentiation to find the derivative of cot x.

d/ dx cotx = d/dx (cosx / sinx).

Quotient rule: d / dx(uv) ={ ( vdu / dx− udv / dx) / v^{2}}

= [(d /dx (cos x) )sin x ]−[ cosx (d/ dx (sin x))] / sin^{2}x -------- (ii)

We can substitute the formulae for the derivative of sin x and cos x given by

d/dx (cos x) = −sin x and d/dx (sin x) = cos x

On putting the above values in equation (ii), we get

d/dx (cot x) = {−sin x sin x − cos x cos x } / sin^{2}x

= - (sin^{2}x + cos^{2}x) / sin^{2}x

= −1 / sin^{2}x

= −(cosec^{2}x)

=> **d/dx (cotx) = −(cosec ^{2}x)**