Solve a^2/a-1-a-1

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

Evaluate

$$\frac{1}{a-1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply -a-1 times \frac{a-1}{a-1}.

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

Since \frac{a^{2}}{a-1} and \frac{\left(-a-1\right)\left(a-1\right)}{a-1} have the same denominator, add them by adding their numerators.

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

Do the multiplications in a^{2}+\left(-a-1\right)\left(a-1\right).

$$\frac{a^{2}-a^{2}+a-a+1}{a-1}$$

Combine like terms in a^{2}-a^{2}+a-a+1.

$$\frac{1}{a-1}$$

Differentiate w.r.t. a

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

Steps Using Chain Rule

To add or subtract expressions, expand them to make their denominators the same. Multiply -a-1 times \frac{a-1}{a-1}.

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

Since \frac{a^{2}}{a-1} and \frac{\left(-a-1\right)\left(a-1\right)}{a-1} have the same denominator, add them by adding their numerators.

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

Do the multiplications in a^{2}+\left(-a-1\right)\left(a-1\right).

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

Combine like terms in a^{2}-a^{2}+a-a+1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a-1})$$

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).

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

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}.

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

Simplify.

$$-a^{0}\left(a^{1}-1\right)^{-2}$$

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

$$-a^{0}\left(a-1\right)^{-2}$$

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

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