Solve d/dxleft(csc(x)right)

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

Evaluate

$$-\frac{\cot(x)}{\sin(x)}$$

Steps Using Derivative Rule for Quotient

Use the definition of cosecant.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{\sin(x)})$$

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.

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

The derivative of the constant 1 is 0, and the derivative of sin(x) is cos(x).

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

Rewrite the quotient as a product of two quotients.

$$\left(-\frac{1}{\sin(x)}\right)\times \frac{\cos(x)}{\sin(x)}$$

Use the definition of cosecant.

$$\left(-\csc(x)\right)\times \frac{\cos(x)}{\sin(x)}$$

Use the definition of cotangent.

$$\left(-\csc(x)\right)\cot(x)$$

Differentiate w.r.t. x

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