Solve 7/3x+24x-1

$$\frac { 7 } { 3 x + 2 } 4 x - 1$$

Evaluate

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

Short Solution Steps

Express \frac{7}{3x+2}\times 4 as a single fraction.

$$\frac{7\times 4}{3x+2}x-1$$

Express \frac{7\times 4}{3x+2}x as a single fraction.

$$\frac{7\times 4x}{3x+2}-1$$

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

$$\frac{7\times 4x}{3x+2}-\frac{3x+2}{3x+2}$$

Since \frac{7\times 4x}{3x+2} and \frac{3x+2}{3x+2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7\times 4x-\left(3x+2\right)}{3x+2}$$

Do the multiplications in 7\times 4x-\left(3x+2\right).

$$\frac{28x-3x-2}{3x+2}$$

Combine like terms in 28x-3x-2.

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

Differentiate w.r.t. x

$$\frac{56}{\left(3x+2\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Express \frac{7}{3x+2}\times 4 as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7\times 4}{3x+2}x-1)$$

Express \frac{7\times 4}{3x+2}x as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7\times 4x}{3x+2}-1)$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7\times 4x}{3x+2}-\frac{3x+2}{3x+2})$$

Since \frac{7\times 4x}{3x+2} and \frac{3x+2}{3x+2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{7\times 4x-\left(3x+2\right)}{3x+2})$$

Do the multiplications in 7\times 4x-\left(3x+2\right).

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{28x-3x-2}{3x+2})$$

Combine like terms in 28x-3x-2.

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

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.

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

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}.

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

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Do the arithmetic.

$$\frac{75x^{1}+50x^{0}-\left(75x^{1}-6x^{0}\right)}{\left(3x^{1}+2\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{75x^{1}+50x^{0}-75x^{1}-\left(-6x^{0}\right)}{\left(3x^{1}+2\right)^{2}}$$

Combine like terms.

$$\frac{\left(75-75\right)x^{1}+\left(50-\left(-6\right)\right)x^{0}}{\left(3x^{1}+2\right)^{2}}$$

Subtract 75 from 75 and -6 from 50.

$$\frac{56x^{0}}{\left(3x^{1}+2\right)^{2}}$$

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

$$\frac{56x^{0}}{\left(3x+2\right)^{2}}$$

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

$$\frac{56\times 1}{\left(3x+2\right)^{2}}$$

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

$$\frac{56}{\left(3x+2\right)^{2}}$$