Solve (x^3/2)^-1=

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

Evaluate

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

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

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

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

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

Raise \frac{1}{2} to the power -1.

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

Differentiate w.r.t. x

$$-\frac{6}{x^{4}}$$

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).

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

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}.

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

Simplify.

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