Solve a/a^2-4/frac{a^2{a+2}}=

$$\frac { \frac { a } { a ^ { 2 } - 4 } } { \frac { a ^ { 2 } } { a + 2 } } =$$

Evaluate

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

Short Solution Steps

Divide \frac{a}{a^{2}-4} by \frac{a^{2}}{a+2} by multiplying \frac{a}{a^{2}-4} by the reciprocal of \frac{a^{2}}{a+2}.

$$\frac{a\left(a+2\right)}{\left(a^{2}-4\right)a^{2}}$$

Cancel out a in both numerator and denominator.

$$\frac{a+2}{a\left(a^{2}-4\right)}$$

Factor the expressions that are not already factored.

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

Cancel out a+2 in both numerator and denominator.

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

Expand the expression.

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

Differentiate w.r.t. a

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

Steps Using Chain Rule

Divide \frac{a}{a^{2}-4} by \frac{a^{2}}{a+2} by multiplying \frac{a}{a^{2}-4} by the reciprocal of \frac{a^{2}}{a+2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a\left(a+2\right)}{\left(a^{2}-4\right)a^{2}})$$

Cancel out a in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a+2}{a\left(a^{2}-4\right)})$$

Factor the expressions that are not already factored in \frac{a+2}{a\left(a^{2}-4\right)}.

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

Cancel out a+2 in both numerator and denominator.

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

Use the distributive property to multiply a by a-2.

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

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).

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

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}.

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

Simplify.

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

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

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

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

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

For any term t, t\times 1=t and 1t=t.

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