Solve 2x+5^2div(3x-2)

$$2x+ { 5 }^{ 2 } \div (3x-2)$$

Evaluate

$$\frac{6x^{2}-4x+25}{3x-2}$$

Short Solution Steps

Calculate 5 to the power of 2 and get 25.

$$2x+\frac{25}{3x-2}$$

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

$$\frac{2x\left(3x-2\right)}{3x-2}+\frac{25}{3x-2}$$

Since \frac{2x\left(3x-2\right)}{3x-2} and \frac{25}{3x-2} have the same denominator, add them by adding their numerators.

$$\frac{2x\left(3x-2\right)+25}{3x-2}$$

Do the multiplications in 2x\left(3x-2\right)+25.

$$\frac{6x^{2}-4x+25}{3x-2}$$

Differentiate w.r.t. x

$$\frac{18x^{2}-24x-67}{\left(3x-2\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Calculate 5 to the power of 2 and get 25.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x+\frac{25}{3x-2})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x\left(3x-2\right)}{3x-2}+\frac{25}{3x-2})$$

Since \frac{2x\left(3x-2\right)}{3x-2} and \frac{25}{3x-2} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x\left(3x-2\right)+25}{3x-2})$$

Do the multiplications in 2x\left(3x-2\right)+25.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{6x^{2}-4x+25}{3x-2})$$

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.

$$\frac{\left(3x^{1}-2\right)\frac{\mathrm{d}}{\mathrm{d}x}(6x^{2}-4x^{1}+25)-\left(6x^{2}-4x^{1}+25\right)\frac{\mathrm{d}}{\mathrm{d}x}(3x^{1}-2)}{\left(3x^{1}-2\right)^{2}}$$

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}.

$$\frac{\left(3x^{1}-2\right)\left(2\times 6x^{2-1}-4x^{1-1}\right)-\left(6x^{2}-4x^{1}+25\right)\times 3x^{1-1}}{\left(3x^{1}-2\right)^{2}}$$

Simplify.

$$\frac{\left(3x^{1}-2\right)\left(12x^{1}-4x^{0}\right)-\left(6x^{2}-4x^{1}+25\right)\times 3x^{0}}{\left(3x^{1}-2\right)^{2}}$$

Multiply 3x^{1}-2 times 12x^{1}-4x^{0}.

$$\frac{3x^{1}\times 12x^{1}+3x^{1}\left(-4\right)x^{0}-2\times 12x^{1}-2\left(-4\right)x^{0}-\left(6x^{2}-4x^{1}+25\right)\times 3x^{0}}{\left(3x^{1}-2\right)^{2}}$$

Multiply 6x^{2}-4x^{1}+25 times 3x^{0}.

$$\frac{3x^{1}\times 12x^{1}+3x^{1}\left(-4\right)x^{0}-2\times 12x^{1}-2\left(-4\right)x^{0}-\left(6x^{2}\times 3x^{0}-4x^{1}\times 3x^{0}+25\times 3x^{0}\right)}{\left(3x^{1}-2\right)^{2}}$$

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

$$\frac{3\times 12x^{1+1}+3\left(-4\right)x^{1}-2\times 12x^{1}-2\left(-4\right)x^{0}-\left(6\times 3x^{2}-4\times 3x^{1}+25\times 3x^{0}\right)}{\left(3x^{1}-2\right)^{2}}$$

Simplify.

$$\frac{36x^{2}-12x^{1}-24x^{1}+8x^{0}-\left(18x^{2}-12x^{1}+75x^{0}\right)}{\left(3x^{1}-2\right)^{2}}$$

Combine like terms.

$$\frac{18x^{2}-24x^{1}-67x^{0}}{\left(3x^{1}-2\right)^{2}}$$

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

$$\frac{18x^{2}-24x-67x^{0}}{\left(3x-2\right)^{2}}$$

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

$$\frac{18x^{2}-24x-67}{\left(3x-2\right)^{2}}$$