Solve d/dxleft(cot(x)right)

$$\frac{d}{d x } \left( \cot ( x ) \right)$$

Evaluate

$$-\frac{1}{\left(\sin(x)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Use the definition of cotangent.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\cos(x)}{\sin(x)})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\sin(x)\frac{\mathrm{d}}{\mathrm{d}x}(\cos(x))-\cos(x)\frac{\mathrm{d}}{\mathrm{d}x}(\sin(x))}{\left(\sin(x)\right)^{2}}$$

The derivative of sin(x) is cos(x), and the derivative of cos(x) is −sin(x).

$$\frac{\sin(x)\left(-\sin(x)\right)-\cos(x)\cos(x)}{\left(\sin(x)\right)^{2}}$$

Simplify.

$$-\frac{\left(\sin(x)\right)^{2}+\left(\cos(x)\right)^{2}}{\left(\sin(x)\right)^{2}}$$

Use the Pythagorean Identity.

$$-\frac{1}{\left(\sin(x)\right)^{2}}$$

Use the definition of cosecant.

$$-\left(\csc(x)\right)^{2}$$

Differentiate w.r.t. x

$$\frac{2\cot(x)}{\left(\sin(x)\right)^{2}}$$