Solve 7x/7x-1-1/1-7x=

$$\frac { 7 x } { 7 x - 1 } - \frac { 1 } { 1 - 7 x } =$$

Evaluate

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

Short Solution Steps

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

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

Since \frac{7x}{7x-1} and \frac{-1}{7x-1} have the same denominator, subtract them by subtracting their numerators.

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

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

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

Differentiate w.r.t. x

$$-\frac{14}{\left(7x-1\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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

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

Since \frac{7x}{7x-1} and \frac{-1}{7x-1} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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.

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

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}.

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

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Do the arithmetic.

$$\frac{49x^{1}-7x^{0}-\left(49x^{1}+7x^{0}\right)}{\left(7x^{1}-1\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{49x^{1}-7x^{0}-49x^{1}-7x^{0}}{\left(7x^{1}-1\right)^{2}}$$

Combine like terms.

$$\frac{\left(49-49\right)x^{1}+\left(-7-7\right)x^{0}}{\left(7x^{1}-1\right)^{2}}$$

Subtract 49 from 49 and 7 from -7.

$$\frac{-14x^{0}}{\left(7x^{1}-1\right)^{2}}$$

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

$$\frac{-14x^{0}}{\left(7x-1\right)^{2}}$$

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

$$\frac{-14}{\left(7x-1\right)^{2}}$$