Solve 6/x+1+5/x+6

$$\frac { 6 } { x + 1 } + \frac { 5 } { x + 6 }$$

Evaluate

$$\frac{11x+41}{\left(x+1\right)\left(x+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+1 and x+6 is \left(x+1\right)\left(x+6\right). Multiply \frac{6}{x+1} times \frac{x+6}{x+6}. Multiply \frac{5}{x+6} times \frac{x+1}{x+1}.

$$\frac{6\left(x+6\right)}{\left(x+1\right)\left(x+6\right)}+\frac{5\left(x+1\right)}{\left(x+1\right)\left(x+6\right)}$$

Since \frac{6\left(x+6\right)}{\left(x+1\right)\left(x+6\right)} and \frac{5\left(x+1\right)}{\left(x+1\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{6\left(x+6\right)+5\left(x+1\right)}{\left(x+1\right)\left(x+6\right)}$$

Do the multiplications in 6\left(x+6\right)+5\left(x+1\right).

$$\frac{6x+36+5x+5}{\left(x+1\right)\left(x+6\right)}$$

Combine like terms in 6x+36+5x+5.

$$\frac{11x+41}{\left(x+1\right)\left(x+6\right)}$$

Expand \left(x+1\right)\left(x+6\right).

$$\frac{11x+41}{x^{2}+7x+6}$$

Differentiate w.r.t. x

$$-\frac{11x^{2}+82x+221}{\left(\left(x+1\right)\left(x+6\right)\right)^{2}}$$

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 x+6 is \left(x+1\right)\left(x+6\right). Multiply \frac{6}{x+1} times \frac{x+6}{x+6}. Multiply \frac{5}{x+6} times \frac{x+1}{x+1}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6\left(x+6\right)}{\left(x+1\right)\left(x+6\right)}+\frac{5\left(x+1\right)}{\left(x+1\right)\left(x+6\right)})$$

Since \frac{6\left(x+6\right)}{\left(x+1\right)\left(x+6\right)} and \frac{5\left(x+1\right)}{\left(x+1\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6\left(x+6\right)+5\left(x+1\right)}{\left(x+1\right)\left(x+6\right)})$$

Do the multiplications in 6\left(x+6\right)+5\left(x+1\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6x+36+5x+5}{\left(x+1\right)\left(x+6\right)})$$

Combine like terms in 6x+36+5x+5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{11x+41}{\left(x+1\right)\left(x+6\right)})$$

Apply the distributive property by multiplying each term of x+1 by each term of x+6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{11x+41}{x^{2}+6x+x+6})$$

Combine 6x and x to get 7x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{11x+41}{x^{2}+7x+6})$$

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.

$$\frac{\left(x^{2}+7x^{1}+6\right)\frac{\mathrm{d}}{\mathrm{d}x}(11x^{1}+41)-\left(11x^{1}+41\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}+7x^{1}+6)}{\left(x^{2}+7x^{1}+6\right)^{2}}$$

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}.

$$\frac{\left(x^{2}+7x^{1}+6\right)\times 11x^{1-1}-\left(11x^{1}+41\right)\left(2x^{2-1}+7x^{1-1}\right)}{\left(x^{2}+7x^{1}+6\right)^{2}}$$

Simplify.

$$\frac{\left(x^{2}+7x^{1}+6\right)\times 11x^{0}-\left(11x^{1}+41\right)\left(2x^{1}+7x^{0}\right)}{\left(x^{2}+7x^{1}+6\right)^{2}}$$

Multiply x^{2}+7x^{1}+6 times 11x^{0}.

$$\frac{x^{2}\times 11x^{0}+7x^{1}\times 11x^{0}+6\times 11x^{0}-\left(11x^{1}+41\right)\left(2x^{1}+7x^{0}\right)}{\left(x^{2}+7x^{1}+6\right)^{2}}$$

Multiply 11x^{1}+41 times 2x^{1}+7x^{0}.

$$\frac{x^{2}\times 11x^{0}+7x^{1}\times 11x^{0}+6\times 11x^{0}-\left(11x^{1}\times 2x^{1}+11x^{1}\times 7x^{0}+41\times 2x^{1}+41\times 7x^{0}\right)}{\left(x^{2}+7x^{1}+6\right)^{2}}$$

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

$$\frac{11x^{2}+7\times 11x^{1}+6\times 11x^{0}-\left(11\times 2x^{1+1}+11\times 7x^{1}+41\times 2x^{1}+41\times 7x^{0}\right)}{\left(x^{2}+7x^{1}+6\right)^{2}}$$

Simplify.

$$\frac{11x^{2}+77x^{1}+66x^{0}-\left(22x^{2}+77x^{1}+82x^{1}+287x^{0}\right)}{\left(x^{2}+7x^{1}+6\right)^{2}}$$

Combine like terms.

$$\frac{-11x^{2}-82x^{1}-221x^{0}}{\left(x^{2}+7x^{1}+6\right)^{2}}$$

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

$$\frac{-11x^{2}-82x-221x^{0}}{\left(x^{2}+7x+6\right)^{2}}$$

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

$$\frac{-11x^{2}-82x-221}{\left(x^{2}+7x+6\right)^{2}}$$