Solve {left({x}^frac{1{2}}right)}^-3

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

Differentiate w.r.t. x

$$-\frac{3}{2x^{\frac{5}{2}}}$$

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).

$$-3\left(\sqrt{x}\right)^{-3-1}\frac{\mathrm{d}}{\mathrm{d}x}(\sqrt{x})$$

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}.

$$-3\left(\sqrt{x}\right)^{-4}\times \frac{1}{2}x^{\frac{1}{2}-1}$$

Simplify.

$$-\frac{3}{2}x^{-\frac{1}{2}}\left(\sqrt{x}\right)^{-4}$$

Evaluate

$$\frac{1}{x^{\frac{3}{2}}}$$