Solve 5+8/4x+3

$$5 + \frac { 8 } { 4 x + 3 }$$

Evaluate

$$\frac{20x+23}{4x+3}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{4x+3}{4x+3}.

$$\frac{5\left(4x+3\right)}{4x+3}+\frac{8}{4x+3}$$

Since \frac{5\left(4x+3\right)}{4x+3} and \frac{8}{4x+3} have the same denominator, add them by adding their numerators.

$$\frac{5\left(4x+3\right)+8}{4x+3}$$

Do the multiplications in 5\left(4x+3\right)+8.

$$\frac{20x+15+8}{4x+3}$$

Combine like terms in 20x+15+8.

$$\frac{20x+23}{4x+3}$$

Differentiate w.r.t. x

$$-\frac{32}{\left(4x+3\right)^{2}}$$

Steps Using Derivative Rule for Quotient

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{4x+3}{4x+3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(4x+3\right)}{4x+3}+\frac{8}{4x+3})$$

Since \frac{5\left(4x+3\right)}{4x+3} and \frac{8}{4x+3} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(4x+3\right)+8}{4x+3})$$

Do the multiplications in 5\left(4x+3\right)+8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{20x+15+8}{4x+3})$$

Combine like terms in 20x+15+8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{20x+23}{4x+3})$$

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.

$$\frac{\left(4x^{1}+3\right)\frac{\mathrm{d}}{\mathrm{d}x}(20x^{1}+23)-\left(20x^{1}+23\right)\frac{\mathrm{d}}{\mathrm{d}x}(4x^{1}+3)}{\left(4x^{1}+3\right)^{2}}$$

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}.

$$\frac{\left(4x^{1}+3\right)\times 20x^{1-1}-\left(20x^{1}+23\right)\times 4x^{1-1}}{\left(4x^{1}+3\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(4x^{1}+3\right)\times 20x^{0}-\left(20x^{1}+23\right)\times 4x^{0}}{\left(4x^{1}+3\right)^{2}}$$

Expand using distributive property.

$$\frac{4x^{1}\times 20x^{0}+3\times 20x^{0}-\left(20x^{1}\times 4x^{0}+23\times 4x^{0}\right)}{\left(4x^{1}+3\right)^{2}}$$

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

$$\frac{4\times 20x^{1}+3\times 20x^{0}-\left(20\times 4x^{1}+23\times 4x^{0}\right)}{\left(4x^{1}+3\right)^{2}}$$

Do the arithmetic.

$$\frac{80x^{1}+60x^{0}-\left(80x^{1}+92x^{0}\right)}{\left(4x^{1}+3\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{80x^{1}+60x^{0}-80x^{1}-92x^{0}}{\left(4x^{1}+3\right)^{2}}$$

Combine like terms.

$$\frac{\left(80-80\right)x^{1}+\left(60-92\right)x^{0}}{\left(4x^{1}+3\right)^{2}}$$

Subtract 80 from 80 and 92 from 60.

$$\frac{-32x^{0}}{\left(4x^{1}+3\right)^{2}}$$

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

$$\frac{-32x^{0}}{\left(4x+3\right)^{2}}$$

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

$$\frac{-32}{\left(4x+3\right)^{2}}$$