Solve (6)4/a^2-4+2/a+2-1/a-2

$$( 6 ) \frac { 4 } { a ^ { 2 } - 4 } + \frac { 2 } { a + 2 } - \frac { 1 } { a - 2 }$$

Evaluate

$$\frac{a+18}{a^{2}-4}$$

Short Solution Steps

Express 6\times \frac{4}{a^{2}-4} as a single fraction.

$$\frac{6\times 4}{a^{2}-4}+\frac{2}{a+2}-\frac{1}{a-2}$$

Factor a^{2}-4.

$$\frac{6\times 4}{\left(a-2\right)\left(a+2\right)}+\frac{2}{a+2}-\frac{1}{a-2}$$

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

$$\frac{6\times 4}{\left(a-2\right)\left(a+2\right)}+\frac{2\left(a-2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{1}{a-2}$$

Since \frac{6\times 4}{\left(a-2\right)\left(a+2\right)} and \frac{2\left(a-2\right)}{\left(a-2\right)\left(a+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{6\times 4+2\left(a-2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{1}{a-2}$$

Do the multiplications in 6\times 4+2\left(a-2\right).

$$\frac{24+2a-4}{\left(a-2\right)\left(a+2\right)}-\frac{1}{a-2}$$

Combine like terms in 24+2a-4.

$$\frac{20+2a}{\left(a-2\right)\left(a+2\right)}-\frac{1}{a-2}$$

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

$$\frac{20+2a}{\left(a-2\right)\left(a+2\right)}-\frac{a+2}{\left(a-2\right)\left(a+2\right)}$$

Since \frac{20+2a}{\left(a-2\right)\left(a+2\right)} and \frac{a+2}{\left(a-2\right)\left(a+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{20+2a-\left(a+2\right)}{\left(a-2\right)\left(a+2\right)}$$

Do the multiplications in 20+2a-\left(a+2\right).

$$\frac{20+2a-a-2}{\left(a-2\right)\left(a+2\right)}$$

Combine like terms in 20+2a-a-2.

$$\frac{18+a}{\left(a-2\right)\left(a+2\right)}$$

Expand \left(a-2\right)\left(a+2\right).

$$\frac{18+a}{a^{2}-4}$$

Differentiate w.r.t. a

$$\frac{-a^{2}-36a-4}{\left(a^{2}-4\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Express 6\times \frac{4}{a^{2}-4} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{6\times 4}{a^{2}-4}+\frac{2}{a+2}-\frac{1}{a-2})$$

Factor a^{2}-4.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{6\times 4}{\left(a-2\right)\left(a+2\right)}+\frac{2}{a+2}-\frac{1}{a-2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{6\times 4}{\left(a-2\right)\left(a+2\right)}+\frac{2\left(a-2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{1}{a-2})$$

Since \frac{6\times 4}{\left(a-2\right)\left(a+2\right)} and \frac{2\left(a-2\right)}{\left(a-2\right)\left(a+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{6\times 4+2\left(a-2\right)}{\left(a-2\right)\left(a+2\right)}-\frac{1}{a-2})$$

Do the multiplications in 6\times 4+2\left(a-2\right).

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{24+2a-4}{\left(a-2\right)\left(a+2\right)}-\frac{1}{a-2})$$

Combine like terms in 24+2a-4.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{20+2a}{\left(a-2\right)\left(a+2\right)}-\frac{1}{a-2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{20+2a}{\left(a-2\right)\left(a+2\right)}-\frac{a+2}{\left(a-2\right)\left(a+2\right)})$$

Since \frac{20+2a}{\left(a-2\right)\left(a+2\right)} and \frac{a+2}{\left(a-2\right)\left(a+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{20+2a-\left(a+2\right)}{\left(a-2\right)\left(a+2\right)})$$

Do the multiplications in 20+2a-\left(a+2\right).

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{20+2a-a-2}{\left(a-2\right)\left(a+2\right)})$$

Combine like terms in 20+2a-a-2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{18+a}{\left(a-2\right)\left(a+2\right)})$$

Consider \left(a-2\right)\left(a+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{18+a}{a^{2}-4})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(a^{2}-4\right)\frac{\mathrm{d}}{\mathrm{d}a}(a^{1}+18)-\left(a^{1}+18\right)\frac{\mathrm{d}}{\mathrm{d}a}(a^{2}-4)}{\left(a^{2}-4\right)^{2}}$$

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}.

$$\frac{\left(a^{2}-4\right)a^{1-1}-\left(a^{1}+18\right)\times 2a^{2-1}}{\left(a^{2}-4\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(a^{2}-4\right)a^{0}-\left(a^{1}+18\right)\times 2a^{1}}{\left(a^{2}-4\right)^{2}}$$

Expand using distributive property.

$$\frac{a^{2}a^{0}-4a^{0}-\left(a^{1}\times 2a^{1}+18\times 2a^{1}\right)}{\left(a^{2}-4\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{a^{2}-4a^{0}-\left(2a^{1+1}+18\times 2a^{1}\right)}{\left(a^{2}-4\right)^{2}}$$

Do the arithmetic.

$$\frac{a^{2}-4a^{0}-\left(2a^{2}+36a^{1}\right)}{\left(a^{2}-4\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{a^{2}-4a^{0}-2a^{2}-36a^{1}}{\left(a^{2}-4\right)^{2}}$$

Combine like terms.

$$\frac{\left(1-2\right)a^{2}-4a^{0}-36a^{1}}{\left(a^{2}-4\right)^{2}}$$

Subtract 2 from 1.

$$\frac{-a^{2}-4a^{0}-36a^{1}}{\left(a^{2}-4\right)^{2}}$$

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

$$\frac{-a^{2}-4a^{0}-36a}{\left(a^{2}-4\right)^{2}}$$

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

$$\frac{-a^{2}-4-36a}{\left(a^{2}-4\right)^{2}}$$