Solve f(x)=csc3/x

$$f ( x ) = \csc \frac { 3 } { x }$$

Differentiate w.r.t. x

$$\frac{3\cot(\frac{3}{x})}{x^{2}\sin(\frac{3}{x})}$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$\left(-\csc(3\times \frac{1}{x})\right)\cot(3\times \frac{1}{x})\frac{\mathrm{d}}{\mathrm{d}x}(3\times \frac{1}{x})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\left(-\csc(3\times \frac{1}{x})\right)\cot(3\times \frac{1}{x})\left(-1\right)\times 3x^{-1-1}$$

Simplify.

$$\left(-\left(-\frac{3}{x^{2}}\right)\right)\csc(3\times \frac{1}{x})\cot(3\times \frac{1}{x})$$

Evaluate

$$\frac{1}{\sin(\frac{3}{x})}$$