Solve 2x/2+4/2x+12

$$\frac{ 2x }{ 2 } + \frac{ 4 }{ 2x+12 }$$

Evaluate

$$\frac{x^{2}+6x+2}{x+6}$$

Short Solution Steps

Cancel out 2 and 2.

$$x+\frac{4}{2x+12}$$

Factor 2x+12.

$$x+\frac{4}{2\left(x+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2\left(x+6\right)}{2\left(x+6\right)}.

$$\frac{x\times 2\left(x+6\right)}{2\left(x+6\right)}+\frac{4}{2\left(x+6\right)}$$

Since \frac{x\times 2\left(x+6\right)}{2\left(x+6\right)} and \frac{4}{2\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{x\times 2\left(x+6\right)+4}{2\left(x+6\right)}$$

Do the multiplications in x\times 2\left(x+6\right)+4.

$$\frac{2x^{2}+12x+4}{2\left(x+6\right)}$$

Factor the expressions that are not already factored in \frac{2x^{2}+12x+4}{2\left(x+6\right)}.

$$\frac{2\left(x-\left(\sqrt{7}-3\right)\right)\left(x-\left(-\sqrt{7}-3\right)\right)}{2\left(x+6\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(x-\left(\sqrt{7}-3\right)\right)\left(x-\left(-\sqrt{7}-3\right)\right)}{x+6}$$

To find the opposite of \sqrt{7}-3, find the opposite of each term.

$$\frac{\left(x-\sqrt{7}-\left(-3\right)\right)\left(x-\left(-\sqrt{7}-3\right)\right)}{x+6}$$

The opposite of -3 is 3.

$$\frac{\left(x-\sqrt{7}+3\right)\left(x-\left(-\sqrt{7}-3\right)\right)}{x+6}$$

To find the opposite of -\sqrt{7}-3, find the opposite of each term.

$$\frac{\left(x-\sqrt{7}+3\right)\left(x-\left(-\sqrt{7}\right)-\left(-3\right)\right)}{x+6}$$

The opposite of -\sqrt{7} is \sqrt{7}.

$$\frac{\left(x-\sqrt{7}+3\right)\left(x+\sqrt{7}-\left(-3\right)\right)}{x+6}$$

The opposite of -3 is 3.

$$\frac{\left(x-\sqrt{7}+3\right)\left(x+\sqrt{7}+3\right)}{x+6}$$

Apply the distributive property by multiplying each term of x-\sqrt{7}+3 by each term of x+\sqrt{7}+3.

$$\frac{x^{2}+x\sqrt{7}+3x-\sqrt{7}x-\left(\sqrt{7}\right)^{2}-3\sqrt{7}+3x+3\sqrt{7}+9}{x+6}$$

Combine x\sqrt{7} and -\sqrt{7}x to get 0.

$$\frac{x^{2}+3x-\left(\sqrt{7}\right)^{2}-3\sqrt{7}+3x+3\sqrt{7}+9}{x+6}$$

The square of \sqrt{7} is 7.

$$\frac{x^{2}+3x-7-3\sqrt{7}+3x+3\sqrt{7}+9}{x+6}$$

Combine 3x and 3x to get 6x.

$$\frac{x^{2}+6x-7-3\sqrt{7}+3\sqrt{7}+9}{x+6}$$

Combine -3\sqrt{7} and 3\sqrt{7} to get 0.

$$\frac{x^{2}+6x-7+9}{x+6}$$

Add -7 and 9 to get 2.

$$\frac{x^{2}+6x+2}{x+6}$$

Differentiate w.r.t. x

$$\frac{x^{2}+12x+34}{\left(x+6\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Cancel out 2 and 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x+\frac{4}{2x+12})$$

Factor 2x+12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x+\frac{4}{2\left(x+6\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2\left(x+6\right)}{2\left(x+6\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x\times 2\left(x+6\right)}{2\left(x+6\right)}+\frac{4}{2\left(x+6\right)})$$

Since \frac{x\times 2\left(x+6\right)}{2\left(x+6\right)} and \frac{4}{2\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x\times 2\left(x+6\right)+4}{2\left(x+6\right)})$$

Do the multiplications in x\times 2\left(x+6\right)+4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}+12x+4}{2\left(x+6\right)})$$

Factor the expressions that are not already factored in \frac{2x^{2}+12x+4}{2\left(x+6\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\left(x-\left(\sqrt{7}-3\right)\right)\left(x-\left(-\sqrt{7}-3\right)\right)}{2\left(x+6\right)})$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-\left(\sqrt{7}-3\right)\right)\left(x-\left(-\sqrt{7}-3\right)\right)}{x+6})$$

To find the opposite of \sqrt{7}-3, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-\sqrt{7}-\left(-3\right)\right)\left(x-\left(-\sqrt{7}-3\right)\right)}{x+6})$$

The opposite of -3 is 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-\sqrt{7}+3\right)\left(x-\left(-\sqrt{7}-3\right)\right)}{x+6})$$

To find the opposite of -\sqrt{7}-3, find the opposite of each term.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-\sqrt{7}+3\right)\left(x-\left(-\sqrt{7}\right)-\left(-3\right)\right)}{x+6})$$

The opposite of -\sqrt{7} is \sqrt{7}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-\sqrt{7}+3\right)\left(x+\sqrt{7}-\left(-3\right)\right)}{x+6})$$

The opposite of -3 is 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-\sqrt{7}+3\right)\left(x+\sqrt{7}+3\right)}{x+6})$$

Apply the distributive property by multiplying each term of x-\sqrt{7}+3 by each term of x+\sqrt{7}+3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+x\sqrt{7}+3x-\sqrt{7}x-\left(\sqrt{7}\right)^{2}-3\sqrt{7}+3x+3\sqrt{7}+9}{x+6})$$

Combine x\sqrt{7} and -\sqrt{7}x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+3x-\left(\sqrt{7}\right)^{2}-3\sqrt{7}+3x+3\sqrt{7}+9}{x+6})$$

The square of \sqrt{7} is 7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+3x-7-3\sqrt{7}+3x+3\sqrt{7}+9}{x+6})$$

Combine 3x and 3x to get 6x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+6x-7-3\sqrt{7}+3\sqrt{7}+9}{x+6})$$

Combine -3\sqrt{7} and 3\sqrt{7} to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+6x-7+9}{x+6})$$

Add -7 and 9 to get 2.

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

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.

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

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}.

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

Simplify.

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

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

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

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

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

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

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

Simplify.

$$\frac{2x^{2}+6x^{1}+12x^{1}+36x^{0}-\left(x^{2}+6x^{1}+2x^{0}\right)}{\left(x^{1}+6\right)^{2}}$$

Combine like terms.

$$\frac{x^{2}+12x^{1}+34x^{0}}{\left(x^{1}+6\right)^{2}}$$

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

$$\frac{x^{2}+12x+34x^{0}}{\left(x+6\right)^{2}}$$

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

$$\frac{x^{2}+12x+34\times 1}{\left(x+6\right)^{2}}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{x^{2}+12x+34}{\left(x+6\right)^{2}}$$