Solve 2x+8+3/3x+12

$$2 x + 8 + \frac { 3 } { 3 x + 12 }$$

Evaluate

$$\frac{2x^{2}+16x+33}{x+4}$$

Short Solution Steps

Factor 3x+12.

$$2x+8+\frac{3}{3\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x+8 times \frac{3\left(x+4\right)}{3\left(x+4\right)}.

$$\frac{\left(2x+8\right)\times 3\left(x+4\right)}{3\left(x+4\right)}+\frac{3}{3\left(x+4\right)}$$

Since \frac{\left(2x+8\right)\times 3\left(x+4\right)}{3\left(x+4\right)} and \frac{3}{3\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x+8\right)\times 3\left(x+4\right)+3}{3\left(x+4\right)}$$

Do the multiplications in \left(2x+8\right)\times 3\left(x+4\right)+3.

$$\frac{6x^{2}+24x+24x+96+3}{3\left(x+4\right)}$$

Combine like terms in 6x^{2}+24x+24x+96+3.

$$\frac{6x^{2}+48x+99}{3\left(x+4\right)}$$

Factor the expressions that are not already factored in \frac{6x^{2}+48x+99}{3\left(x+4\right)}.

$$\frac{3\left(2x^{2}+16x+33\right)}{3\left(x+4\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{2x^{2}+16x+33}{x+4}$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Quotient

Factor 3x+12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x+8+\frac{3}{3\left(x+4\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2x+8 times \frac{3\left(x+4\right)}{3\left(x+4\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(2x+8\right)\times 3\left(x+4\right)}{3\left(x+4\right)}+\frac{3}{3\left(x+4\right)})$$

Since \frac{\left(2x+8\right)\times 3\left(x+4\right)}{3\left(x+4\right)} and \frac{3}{3\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(2x+8\right)\times 3\left(x+4\right)+3}{3\left(x+4\right)})$$

Do the multiplications in \left(2x+8\right)\times 3\left(x+4\right)+3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6x^{2}+24x+24x+96+3}{3\left(x+4\right)})$$

Combine like terms in 6x^{2}+24x+24x+96+3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6x^{2}+48x+99}{3\left(x+4\right)})$$

Factor the expressions that are not already factored in \frac{6x^{2}+48x+99}{3\left(x+4\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3\left(2x^{2}+16x+33\right)}{3\left(x+4\right)})$$

Cancel out 3 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x^{2}+16x+33}{x+4})$$

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.

$$\frac{\left(x^{1}+4\right)\frac{\mathrm{d}}{\mathrm{d}x}(2x^{2}+16x^{1}+33)-\left(2x^{2}+16x^{1}+33\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}+4)}{\left(x^{1}+4\right)^{2}}$$

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}.

$$\frac{\left(x^{1}+4\right)\left(2\times 2x^{2-1}+16x^{1-1}\right)-\left(2x^{2}+16x^{1}+33\right)x^{1-1}}{\left(x^{1}+4\right)^{2}}$$

Simplify.

$$\frac{\left(x^{1}+4\right)\left(4x^{1}+16x^{0}\right)-\left(2x^{2}+16x^{1}+33\right)x^{0}}{\left(x^{1}+4\right)^{2}}$$

Multiply x^{1}+4 times 4x^{1}+16x^{0}.

$$\frac{x^{1}\times 4x^{1}+x^{1}\times 16x^{0}+4\times 4x^{1}+4\times 16x^{0}-\left(2x^{2}+16x^{1}+33\right)x^{0}}{\left(x^{1}+4\right)^{2}}$$

Multiply 2x^{2}+16x^{1}+33 times x^{0}.

$$\frac{x^{1}\times 4x^{1}+x^{1}\times 16x^{0}+4\times 4x^{1}+4\times 16x^{0}-\left(2x^{2}x^{0}+16x^{1}x^{0}+33x^{0}\right)}{\left(x^{1}+4\right)^{2}}$$

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

$$\frac{4x^{1+1}+16x^{1}+4\times 4x^{1}+4\times 16x^{0}-\left(2x^{2}+16x^{1}+33x^{0}\right)}{\left(x^{1}+4\right)^{2}}$$

Simplify.

$$\frac{4x^{2}+16x^{1}+16x^{1}+64x^{0}-\left(2x^{2}+16x^{1}+33x^{0}\right)}{\left(x^{1}+4\right)^{2}}$$

Combine like terms.

$$\frac{2x^{2}+16x^{1}+31x^{0}}{\left(x^{1}+4\right)^{2}}$$

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

$$\frac{2x^{2}+16x+31x^{0}}{\left(x+4\right)^{2}}$$

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

$$\frac{2x^{2}+16x+31\times 1}{\left(x+4\right)^{2}}$$

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

$$\frac{2x^{2}+16x+31}{\left(x+4\right)^{2}}$$