Solve 1/2x-1+1/2x+1+x/4x^2-1

$$\frac { 1 } { 2 x - 1 } + \frac { 1 } { 2 x + 1 } + \frac { x } { 4 x ^ { 2 } - 1 }$$

Evaluate

$$\frac{5x}{4x^{2}-1}$$

Short Solution Steps

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

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

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

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

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

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

Factor 4x^{2}-1.

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

Since \frac{4x}{\left(2x-1\right)\left(2x+1\right)} and \frac{x}{\left(2x-1\right)\left(2x+1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in 4x+x.

$$\frac{5x}{\left(2x-1\right)\left(2x+1\right)}$$

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

$$\frac{5x}{4x^{2}-1}$$

Differentiate w.r.t. x

$$-\frac{5\left(4x^{2}+1\right)}{\left(4x^{2}-1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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

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

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

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

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

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

Factor 4x^{2}-1.

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

Since \frac{4x}{\left(2x-1\right)\left(2x+1\right)} and \frac{x}{\left(2x-1\right)\left(2x+1\right)} have the same denominator, add them by adding their numerators.

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

Combine like terms in 4x+x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x}{\left(2x-1\right)\left(2x+1\right)})$$

Consider \left(2x-1\right)\left(2x+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}. Square 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x}{\left(2x\right)^{2}-1})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x}{2^{2}x^{2}-1})$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x}{4x^{2}-1})$$

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.

$$\frac{\left(4x^{2}-1\right)\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1})-5x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(4x^{2}-1)}{\left(4x^{2}-1\right)^{2}}$$

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}.

$$\frac{\left(4x^{2}-1\right)\times 5x^{1-1}-5x^{1}\times 2\times 4x^{2-1}}{\left(4x^{2}-1\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(4x^{2}-1\right)\times 5x^{0}-5x^{1}\times 8x^{1}}{\left(4x^{2}-1\right)^{2}}$$

Expand using distributive property.

$$\frac{4x^{2}\times 5x^{0}-5x^{0}-5x^{1}\times 8x^{1}}{\left(4x^{2}-1\right)^{2}}$$

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

$$\frac{4\times 5x^{2}-5x^{0}-5\times 8x^{1+1}}{\left(4x^{2}-1\right)^{2}}$$

Do the arithmetic.

$$\frac{20x^{2}-5x^{0}-40x^{2}}{\left(4x^{2}-1\right)^{2}}$$

Combine like terms.

$$\frac{\left(20-40\right)x^{2}-5x^{0}}{\left(4x^{2}-1\right)^{2}}$$

Subtract 40 from 20.

$$\frac{-20x^{2}-5x^{0}}{\left(4x^{2}-1\right)^{2}}$$

Factor out 5.

$$\frac{5\left(-4x^{2}-x^{0}\right)}{\left(4x^{2}-1\right)^{2}}$$

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

$$\frac{5\left(-4x^{2}-1\right)}{\left(4x^{2}-1\right)^{2}}$$