Solve 42a/a^2-4-1/a-2

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

Evaluate

$$\frac{7a-2}{a^{2}-4}$$

Short Solution Steps

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

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

Factor a^{2}-4.

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

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}.

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

Since \frac{4\times 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.

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

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

$$\frac{8a-a-2}{\left(a-2\right)\left(a+2\right)}$$

Combine like terms in 8a-a-2.

$$\frac{7a-2}{\left(a-2\right)\left(a+2\right)}$$

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

$$\frac{7a-2}{a^{2}-4}$$

Differentiate w.r.t. a

$$\frac{-7a^{2}+4a-28}{\left(a^{2}-4\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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

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

Factor a^{2}-4.

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

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}.

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

Since \frac{4\times 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.

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

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{8a-a-2}{\left(a-2\right)\left(a+2\right)})$$

Combine like terms in 8a-a-2.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{7a-2}{\left(a-2\right)\left(a+2\right)})$$

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.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{7a-2}{a^{2}-4})$$

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.

$$\frac{\left(a^{2}-4\right)\frac{\mathrm{d}}{\mathrm{d}a}(7a^{1}-2)-\left(7a^{1}-2\right)\frac{\mathrm{d}}{\mathrm{d}a}(a^{2}-4)}{\left(a^{2}-4\right)^{2}}$$

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}.

$$\frac{\left(a^{2}-4\right)\times 7a^{1-1}-\left(7a^{1}-2\right)\times 2a^{2-1}}{\left(a^{2}-4\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(a^{2}-4\right)\times 7a^{0}-\left(7a^{1}-2\right)\times 2a^{1}}{\left(a^{2}-4\right)^{2}}$$

Expand using distributive property.

$$\frac{a^{2}\times 7a^{0}-4\times 7a^{0}-\left(7a^{1}\times 2a^{1}-2\times 2a^{1}\right)}{\left(a^{2}-4\right)^{2}}$$

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

$$\frac{7a^{2}-4\times 7a^{0}-\left(7\times 2a^{1+1}-2\times 2a^{1}\right)}{\left(a^{2}-4\right)^{2}}$$

Do the arithmetic.

$$\frac{7a^{2}-28a^{0}-\left(14a^{2}-4a^{1}\right)}{\left(a^{2}-4\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{7a^{2}-28a^{0}-14a^{2}-\left(-4a^{1}\right)}{\left(a^{2}-4\right)^{2}}$$

Combine like terms.

$$\frac{\left(7-14\right)a^{2}-28a^{0}-\left(-4a^{1}\right)}{\left(a^{2}-4\right)^{2}}$$

Subtract 14 from 7.

$$\frac{-7a^{2}-28a^{0}-\left(-4a^{1}\right)}{\left(a^{2}-4\right)^{2}}$$

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

$$\frac{-7a^{2}-28a^{0}-\left(-4a\right)}{\left(a^{2}-4\right)^{2}}$$

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

$$\frac{-7a^{2}-28-\left(-4a\right)}{\left(a^{2}-4\right)^{2}}$$