Solve 5/x-1-4/4x

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and 4x is 4x\left(x-1\right). Multiply \frac{5}{x-1} times \frac{4x}{4x}. Multiply \frac{4}{4x} times \frac{x-1}{x-1}.

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

Since \frac{5\times 4x}{4x\left(x-1\right)} and \frac{4\left(x-1\right)}{4x\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 5\times 4x-4\left(x-1\right).

$$\frac{20x-4x+4}{4x\left(x-1\right)}$$

Combine like terms in 20x-4x+4.

$$\frac{16x+4}{4x\left(x-1\right)}$$

Factor the expressions that are not already factored in \frac{16x+4}{4x\left(x-1\right)}.

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

Cancel out 4 in both numerator and denominator.

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

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

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

Differentiate w.r.t. x

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

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 4x is 4x\left(x-1\right). Multiply \frac{5}{x-1} times \frac{4x}{4x}. Multiply \frac{4}{4x} times \frac{x-1}{x-1}.

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

Since \frac{5\times 4x}{4x\left(x-1\right)} and \frac{4\left(x-1\right)}{4x\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 5\times 4x-4\left(x-1\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{20x-4x+4}{4x\left(x-1\right)})$$

Combine like terms in 20x-4x+4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{16x+4}{4x\left(x-1\right)})$$

Factor the expressions that are not already factored in \frac{16x+4}{4x\left(x-1\right)}.

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

Cancel out 4 in both numerator and denominator.

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

Use the distributive property to multiply x by x-1.

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

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.

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

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}.

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

Simplify.

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

Multiply x^{2}-x^{1} times 4x^{0}.

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

Multiply 4x^{1}+1 times 2x^{1}-x^{0}.

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

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

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

Simplify.

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

Combine like terms.

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

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

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

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

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