Solve 3x+2+1/3x+2

$$3 x + 2 + \frac { 1 } { 3 x + 2 }$$

Evaluate

$$\frac{9x^{2}+12x+5}{3x+2}$$

Short Solution Steps

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

$$\frac{\left(3x+2\right)\left(3x+2\right)}{3x+2}+\frac{1}{3x+2}$$

Since \frac{\left(3x+2\right)\left(3x+2\right)}{3x+2} and \frac{1}{3x+2} have the same denominator, add them by adding their numerators.

$$\frac{\left(3x+2\right)\left(3x+2\right)+1}{3x+2}$$

Do the multiplications in \left(3x+2\right)\left(3x+2\right)+1.

$$\frac{9x^{2}+6x+6x+4+1}{3x+2}$$

Combine like terms in 9x^{2}+6x+6x+4+1.

$$\frac{9x^{2}+12x+5}{3x+2}$$

Differentiate w.r.t. x

$$\frac{9\left(x+1\right)\left(3x+1\right)}{\left(3x+2\right)^{2}}$$

Steps Using Derivative Rule for Quotient

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(3x+2\right)\left(3x+2\right)}{3x+2}+\frac{1}{3x+2})$$

Since \frac{\left(3x+2\right)\left(3x+2\right)}{3x+2} and \frac{1}{3x+2} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(3x+2\right)\left(3x+2\right)+1}{3x+2})$$

Do the multiplications in \left(3x+2\right)\left(3x+2\right)+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9x^{2}+6x+6x+4+1}{3x+2})$$

Combine like terms in 9x^{2}+6x+6x+4+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{9x^{2}+12x+5}{3x+2})$$

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.

$$\frac{\left(3x^{1}+2\right)\frac{\mathrm{d}}{\mathrm{d}x}(9x^{2}+12x^{1}+5)-\left(9x^{2}+12x^{1}+5\right)\frac{\mathrm{d}}{\mathrm{d}x}(3x^{1}+2)}{\left(3x^{1}+2\right)^{2}}$$

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}.

$$\frac{\left(3x^{1}+2\right)\left(2\times 9x^{2-1}+12x^{1-1}\right)-\left(9x^{2}+12x^{1}+5\right)\times 3x^{1-1}}{\left(3x^{1}+2\right)^{2}}$$

Simplify.

$$\frac{\left(3x^{1}+2\right)\left(18x^{1}+12x^{0}\right)-\left(9x^{2}+12x^{1}+5\right)\times 3x^{0}}{\left(3x^{1}+2\right)^{2}}$$

Multiply 3x^{1}+2 times 18x^{1}+12x^{0}.

$$\frac{3x^{1}\times 18x^{1}+3x^{1}\times 12x^{0}+2\times 18x^{1}+2\times 12x^{0}-\left(9x^{2}+12x^{1}+5\right)\times 3x^{0}}{\left(3x^{1}+2\right)^{2}}$$

Multiply 9x^{2}+12x^{1}+5 times 3x^{0}.

$$\frac{3x^{1}\times 18x^{1}+3x^{1}\times 12x^{0}+2\times 18x^{1}+2\times 12x^{0}-\left(9x^{2}\times 3x^{0}+12x^{1}\times 3x^{0}+5\times 3x^{0}\right)}{\left(3x^{1}+2\right)^{2}}$$

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

$$\frac{3\times 18x^{1+1}+3\times 12x^{1}+2\times 18x^{1}+2\times 12x^{0}-\left(9\times 3x^{2}+12\times 3x^{1}+5\times 3x^{0}\right)}{\left(3x^{1}+2\right)^{2}}$$

Simplify.

$$\frac{54x^{2}+36x^{1}+36x^{1}+24x^{0}-\left(27x^{2}+36x^{1}+15x^{0}\right)}{\left(3x^{1}+2\right)^{2}}$$

Combine like terms.

$$\frac{27x^{2}+36x^{1}+9x^{0}}{\left(3x^{1}+2\right)^{2}}$$

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

$$\frac{27x^{2}+36x+9x^{0}}{\left(3x+2\right)^{2}}$$

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

$$\frac{27x^{2}+36x+9\times 1}{\left(3x+2\right)^{2}}$$

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

$$\frac{27x^{2}+36x+9}{\left(3x+2\right)^{2}}$$