Solve x/3x-23x+1

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

Evaluate

$$\frac{3x^{2}+3x-2}{3x-2}$$

Short Solution Steps

Express \frac{x}{3x-2}\times 3 as a single fraction.

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

Express \frac{x\times 3}{3x-2}x as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3x-2}{3x-2}.

$$\frac{x\times 3x}{3x-2}+\frac{3x-2}{3x-2}$$

Since \frac{x\times 3x}{3x-2} and \frac{3x-2}{3x-2} have the same denominator, add them by adding their numerators.

$$\frac{x\times 3x+3x-2}{3x-2}$$

Do the multiplications in x\times 3x+3x-2.

$$\frac{3x^{2}+3x-2}{3x-2}$$

Differentiate w.r.t. x

$$\frac{3x\left(3x-4\right)}{\left(3x-2\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Express \frac{x}{3x-2}\times 3 as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x\times 3}{3x-2}x+1)$$

Express \frac{x\times 3}{3x-2}x as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x\times 3x}{3x-2}+1)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{3x-2}{3x-2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x\times 3x}{3x-2}+\frac{3x-2}{3x-2})$$

Since \frac{x\times 3x}{3x-2} and \frac{3x-2}{3x-2} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x\times 3x+3x-2}{3x-2})$$

Do the multiplications in x\times 3x+3x-2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3x^{2}+3x-2}{3x-2})$$

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.

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

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}.

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

Simplify.

$$\frac{\left(3x^{1}-2\right)\left(6x^{1}+3x^{0}\right)-\left(3x^{2}+3x^{1}-2\right)\times 3x^{0}}{\left(3x^{1}-2\right)^{2}}$$

Multiply 3x^{1}-2 times 6x^{1}+3x^{0}.

$$\frac{3x^{1}\times 6x^{1}+3x^{1}\times 3x^{0}-2\times 6x^{1}-2\times 3x^{0}-\left(3x^{2}+3x^{1}-2\right)\times 3x^{0}}{\left(3x^{1}-2\right)^{2}}$$

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

$$\frac{3x^{1}\times 6x^{1}+3x^{1}\times 3x^{0}-2\times 6x^{1}-2\times 3x^{0}-\left(3x^{2}\times 3x^{0}+3x^{1}\times 3x^{0}-2\times 3x^{0}\right)}{\left(3x^{1}-2\right)^{2}}$$

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

$$\frac{3\times 6x^{1+1}+3\times 3x^{1}-2\times 6x^{1}-2\times 3x^{0}-\left(3\times 3x^{2}+3\times 3x^{1}-2\times 3x^{0}\right)}{\left(3x^{1}-2\right)^{2}}$$

Simplify.

$$\frac{18x^{2}+9x^{1}-12x^{1}-6x^{0}-\left(9x^{2}+9x^{1}-6x^{0}\right)}{\left(3x^{1}-2\right)^{2}}$$

Combine like terms.

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

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

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