Solve 5/(x+1)-2/17-3

$$\frac { 5 } { ( x + 1 ) } - \frac { 2 } { 17 - 3 }$$

Evaluate

$$\frac{34-x}{7\left(x+1\right)}$$

Short Solution Steps

Subtract 3 from 17 to get 14.

$$\frac{5}{x+1}-\frac{2}{14}$$

Reduce the fraction \frac{2}{14} to lowest terms by extracting and canceling out 2.

$$\frac{5}{x+1}-\frac{1}{7}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and 7 is 7\left(x+1\right). Multiply \frac{5}{x+1} times \frac{7}{7}. Multiply \frac{1}{7} times \frac{x+1}{x+1}.

$$\frac{5\times 7}{7\left(x+1\right)}-\frac{x+1}{7\left(x+1\right)}$$

Since \frac{5\times 7}{7\left(x+1\right)} and \frac{x+1}{7\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\times 7-\left(x+1\right)}{7\left(x+1\right)}$$

Do the multiplications in 5\times 7-\left(x+1\right).

$$\frac{35-x-1}{7\left(x+1\right)}$$

Combine like terms in 35-x-1.

$$\frac{34-x}{7\left(x+1\right)}$$

Expand 7\left(x+1\right).

$$\frac{34-x}{7x+7}$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Quotient

Subtract 3 from 17 to get 14.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5}{x+1}-\frac{2}{14})$$

Reduce the fraction \frac{2}{14} to lowest terms by extracting and canceling out 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5}{x+1}-\frac{1}{7})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and 7 is 7\left(x+1\right). Multiply \frac{5}{x+1} times \frac{7}{7}. Multiply \frac{1}{7} times \frac{x+1}{x+1}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\times 7}{7\left(x+1\right)}-\frac{x+1}{7\left(x+1\right)})$$

Since \frac{5\times 7}{7\left(x+1\right)} and \frac{x+1}{7\left(x+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\times 7-\left(x+1\right)}{7\left(x+1\right)})$$

Do the multiplications in 5\times 7-\left(x+1\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{35-x-1}{7\left(x+1\right)})$$

Combine like terms in 35-x-1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{34-x}{7\left(x+1\right)})$$

Use the distributive property to multiply 7 by x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{34-x}{7x+7})$$

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.

$$\frac{\left(7x^{1}+7\right)\frac{\mathrm{d}}{\mathrm{d}x}(-x^{1}+34)-\left(-x^{1}+34\right)\frac{\mathrm{d}}{\mathrm{d}x}(7x^{1}+7)}{\left(7x^{1}+7\right)^{2}}$$

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}.

$$\frac{\left(7x^{1}+7\right)\left(-1\right)x^{1-1}-\left(-x^{1}+34\right)\times 7x^{1-1}}{\left(7x^{1}+7\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(7x^{1}+7\right)\left(-1\right)x^{0}-\left(-x^{1}+34\right)\times 7x^{0}}{\left(7x^{1}+7\right)^{2}}$$

Expand using distributive property.

$$\frac{7x^{1}\left(-1\right)x^{0}+7\left(-1\right)x^{0}-\left(-x^{1}\times 7x^{0}+34\times 7x^{0}\right)}{\left(7x^{1}+7\right)^{2}}$$

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

$$\frac{7\left(-1\right)x^{1}+7\left(-1\right)x^{0}-\left(-7x^{1}+34\times 7x^{0}\right)}{\left(7x^{1}+7\right)^{2}}$$

Do the arithmetic.

$$\frac{-7x^{1}-7x^{0}-\left(-7x^{1}+238x^{0}\right)}{\left(7x^{1}+7\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{-7x^{1}-7x^{0}-\left(-7x^{1}\right)-238x^{0}}{\left(7x^{1}+7\right)^{2}}$$

Combine like terms.

$$\frac{\left(-7-\left(-7\right)\right)x^{1}+\left(-7-238\right)x^{0}}{\left(7x^{1}+7\right)^{2}}$$

Subtract -7 from -7 and 238 from -7.

$$\frac{-245x^{0}}{\left(7x^{1}+7\right)^{2}}$$

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

$$\frac{-245x^{0}}{\left(7x+7\right)^{2}}$$

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

$$\frac{-245}{\left(7x+7\right)^{2}}$$