Solve -1/x^-2/frac{1{x}}

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

Evaluate

$$-x^{3}$$

Solution Steps

Divide -\frac{1}{x^{-2}} by \frac{1}{x} by multiplying -\frac{1}{x^{-2}} by the reciprocal of \frac{1}{x}.

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

Express \left(-\frac{1}{x^{-2}}\right)x as a single fraction.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$-x^{3}$$

Differentiate w.r.t. x

$$-3x^{2}$$

Steps Using Definition of a Derivative

Divide -\frac{1}{x^{-2}} by \frac{1}{x} by multiplying -\frac{1}{x^{-2}} by the reciprocal of \frac{1}{x}.

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

Express \left(-\frac{1}{x^{-2}}\right)x as a single fraction.

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

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

The derivative of ax^{n} is nax^{n-1}.

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

Multiply 3 times -1.

$$-3x^{3-1}$$

Subtract 1 from 3.

$$-3x^{2}$$