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

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

Evaluate

$$\frac{1}{x^{6}}$$

Steps for Raising Quotient to a Power

Use the rules of exponents to simplify the expression.

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

To raise the quotient of two numbers to a power, raise each number to the power and then divide.

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

To raise a power to another power, multiply the exponents.

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

Multiply -2 times -3.

$$\frac{1}{x^{6}}$$

Differentiate w.r.t. x

$$-\frac{6}{x^{7}}$$

Steps Using Chain Rule

To raise \frac{1}{x^{-2}} to a power, raise both numerator and denominator to the power and then divide.

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

To raise a power to another power, multiply the exponents. Multiply -2 and -3 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1^{-3}}{x^{6}})$$

Calculate 1 to the power of -3 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{6}})$$

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).

$$-\left(x^{6}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{6})$$

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}.

$$-\left(x^{6}\right)^{-2}\times 6x^{6-1}$$

Simplify.

$$-6x^{5}\left(x^{6}\right)^{-2}$$