Solve 1/2a+1+1/2a-1

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

Evaluate

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

Short Solution Steps

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

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

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

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

Combine like terms in 2a-1+2a+1.

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

Expand \left(2a-1\right)\left(2a+1\right).

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

Differentiate w.r.t. a

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

Steps Using Derivative Rule for Quotient

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

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

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

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

Combine like terms in 2a-1+2a+1.

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

Consider \left(2a-1\right)\left(2a+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(2a\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

Calculate 1 to the power of 2 and get 1.

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

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.

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

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}.

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

Do the arithmetic.

$$\frac{\left(4a^{2}-1\right)\times 4a^{0}-4a^{1}\times 8a^{1}}{\left(4a^{2}-1\right)^{2}}$$

Expand using distributive property.

$$\frac{4a^{2}\times 4a^{0}-4a^{0}-4a^{1}\times 8a^{1}}{\left(4a^{2}-1\right)^{2}}$$

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

$$\frac{4\times 4a^{2}-4a^{0}-4\times 8a^{1+1}}{\left(4a^{2}-1\right)^{2}}$$

Do the arithmetic.

$$\frac{16a^{2}-4a^{0}-32a^{2}}{\left(4a^{2}-1\right)^{2}}$$

Combine like terms.

$$\frac{\left(16-32\right)a^{2}-4a^{0}}{\left(4a^{2}-1\right)^{2}}$$

Subtract 32 from 16.

$$\frac{-16a^{2}-4a^{0}}{\left(4a^{2}-1\right)^{2}}$$

Factor out 4.

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

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

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