Solve 1/x-1+1/2x+1

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

Evaluate

$$\frac{3x}{\left(x-1\right)\left(2x+1\right)}$$

Short Solution Steps

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

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

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

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

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

$$\frac{3x}{\left(x-1\right)\left(2x+1\right)}$$

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

$$\frac{3x}{2x^{2}-x-1}$$

Differentiate w.r.t. x

$$-\frac{3\left(2x^{2}+1\right)}{\left(\left(x-1\right)\left(2x+1\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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

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

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x}{\left(x-1\right)\left(2x+1\right)})$$

Apply the distributive property by multiplying each term of x-1 by each term of 2x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x}{2x^{2}+x-2x-1})$$

Combine x and -2x to get -x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x}{2x^{2}-x-1})$$

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.

$$\frac{\left(2x^{2}-x^{1}-1\right)\frac{\mathrm{d}}{\mathrm{d}x}(3x^{1})-3x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(2x^{2}-x^{1}-1)}{\left(2x^{2}-x^{1}-1\right)^{2}}$$

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}.

$$\frac{\left(2x^{2}-x^{1}-1\right)\times 3x^{1-1}-3x^{1}\left(2\times 2x^{2-1}-x^{1-1}\right)}{\left(2x^{2}-x^{1}-1\right)^{2}}$$

Simplify.

$$\frac{\left(2x^{2}-x^{1}-1\right)\times 3x^{0}-3x^{1}\left(4x^{1}-x^{0}\right)}{\left(2x^{2}-x^{1}-1\right)^{2}}$$

Multiply 2x^{2}-x^{1}-1 times 3x^{0}.

$$\frac{2x^{2}\times 3x^{0}-x^{1}\times 3x^{0}-3x^{0}-3x^{1}\left(4x^{1}-x^{0}\right)}{\left(2x^{2}-x^{1}-1\right)^{2}}$$

Multiply 3x^{1} times 4x^{1}-x^{0}.

$$\frac{2x^{2}\times 3x^{0}-x^{1}\times 3x^{0}-3x^{0}-\left(3x^{1}\times 4x^{1}+3x^{1}\left(-1\right)x^{0}\right)}{\left(2x^{2}-x^{1}-1\right)^{2}}$$

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

$$\frac{2\times 3x^{2}-3x^{1}-3x^{0}-\left(3\times 4x^{1+1}+3\left(-1\right)x^{1}\right)}{\left(2x^{2}-x^{1}-1\right)^{2}}$$

Simplify.

$$\frac{6x^{2}-3x^{1}-3x^{0}-\left(12x^{2}-3x^{1}\right)}{\left(2x^{2}-x^{1}-1\right)^{2}}$$

Combine like terms.

$$\frac{-6x^{2}-3x^{0}}{\left(2x^{2}-x^{1}-1\right)^{2}}$$

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

$$\frac{-6x^{2}-3x^{0}}{\left(2x^{2}-x-1\right)^{2}}$$

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

$$\frac{-6x^{2}-3}{\left(2x^{2}-x-1\right)^{2}}$$