Solve 1/a-2-4/a^2-4

$$\frac { 1 } { a - 2 } - \frac { 4 } { a ^ { 2 } - 4 }$$

Evaluate

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

Short Solution Steps

Factor a^{2}-4.

$$\frac{1}{a-2}-\frac{4}{\left(a-2\right)\left(a+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-2 and \left(a-2\right)\left(a+2\right) is \left(a-2\right)\left(a+2\right). Multiply \frac{1}{a-2} times \frac{a+2}{a+2}.

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

Since \frac{a+2}{\left(a-2\right)\left(a+2\right)} and \frac{4}{\left(a-2\right)\left(a+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in a+2-4.

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

Cancel out a-2 in both numerator and denominator.

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

Differentiate w.r.t. a

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

Steps Using Chain Rule

Factor a^{2}-4.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a-2}-\frac{4}{\left(a-2\right)\left(a+2\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-2 and \left(a-2\right)\left(a+2\right) is \left(a-2\right)\left(a+2\right). Multiply \frac{1}{a-2} times \frac{a+2}{a+2}.

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

Since \frac{a+2}{\left(a-2\right)\left(a+2\right)} and \frac{4}{\left(a-2\right)\left(a+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Combine like terms in a+2-4.

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

Cancel out a-2 in both numerator and denominator.

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

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).

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

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}.

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

Simplify.

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

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

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

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

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