Solve 1/x^3/x^-5

$$\frac { \frac { 1 } { x ^ { 3 } } } { x ^ { - 5 } }$$

Evaluate

$$x^{2}$$

Steps for the Product of Two Exponentials

Use the rules of exponents to simplify the expression.

$$\frac{1}{x^{3}}\times \frac{1}{x^{-5}}$$

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

$$x^{3\left(-1\right)}x^{-5\left(-1\right)}$$

Multiply 3 times -1.

$$x^{-3}x^{-5\left(-1\right)}$$

Multiply -5 times -1.

$$x^{-3}x^{5}$$

To multiply powers of the same base, add their exponents.

$$x^{-3+5}$$

Add the exponents -3 and 5.

$$x^{2}$$

Differentiate w.r.t. x

$$2x$$

Steps Using Chain Rule

Express \frac{\frac{1}{x^{3}}}{x^{-5}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{3}x^{-5}})$$

To multiply powers of the same base, add their exponents. Add 3 and -5 to get -2.

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

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).

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

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}.

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

Simplify.

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