Solve 1/a^2-a:1/a^2-a^3

$$\frac { 1 } { a ^ { 2 } - a } : \frac { 1 } { a ^ { 2 } - a ^ { 3 } }$$

Evaluate

$$-a$$

Short Solution Steps

Divide \frac{1}{a^{2}-a} by \frac{1}{a^{2}-a^{3}} by multiplying \frac{1}{a^{2}-a} by the reciprocal of \frac{1}{a^{2}-a^{3}}.

$$\frac{a^{2}-a^{3}}{a^{2}-a}$$

Factor the expressions that are not already factored.

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

Extract the negative sign in 1-a.

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

Cancel out a\left(a-1\right) in both numerator and denominator.

$$-a$$

Differentiate w.r.t. a

$$-1$$

Steps Using Definition of a Derivative

Divide \frac{1}{a^{2}-a} by \frac{1}{a^{2}-a^{3}} by multiplying \frac{1}{a^{2}-a} by the reciprocal of \frac{1}{a^{2}-a^{3}}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{2}-a^{3}}{a^{2}-a})$$

Factor the expressions that are not already factored in \frac{a^{2}-a^{3}}{a^{2}-a}.

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

Extract the negative sign in 1-a.

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

Cancel out a\left(a-1\right) in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-a)$$

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

$$-a^{1-1}$$

Subtract 1 from 1.

$$-a^{0}$$

For any term t except 0, t^{0}=1.

$$-1$$