Solve 5x:8x^2/x+1

$$5 x : \frac { 8 x ^ { 2 } } { x + 1 }$$

Evaluate

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

Short Solution Steps

Divide 5x by \frac{8x^{2}}{x+1} by multiplying 5x by the reciprocal of \frac{8x^{2}}{x+1}.

$$\frac{5x\left(x+1\right)}{8x^{2}}$$

Cancel out x in both numerator and denominator.

$$\frac{5\left(x+1\right)}{8x}$$

Use the distributive property to multiply 5 by x+1.

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

Differentiate w.r.t. x

$$-\frac{5}{8x^{2}}$$

Steps Using Derivative Rule for Quotient

Divide 5x by \frac{8x^{2}}{x+1} by multiplying 5x by the reciprocal of \frac{8x^{2}}{x+1}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x\left(x+1\right)}{8x^{2}})$$

Cancel out x in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5\left(x+1\right)}{8x})$$

Use the distributive property to multiply 5 by x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{5x+5}{8x})$$

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.

$$\frac{8x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(5x^{1}+5)-\left(5x^{1}+5\right)\frac{\mathrm{d}}{\mathrm{d}x}(8x^{1})}{\left(8x^{1}\right)^{2}}$$

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}.

$$\frac{8x^{1}\times 5x^{1-1}-\left(5x^{1}+5\right)\times 8x^{1-1}}{\left(8x^{1}\right)^{2}}$$

Do the arithmetic.

$$\frac{8x^{1}\times 5x^{0}-\left(5x^{1}+5\right)\times 8x^{0}}{\left(8x^{1}\right)^{2}}$$

Expand using distributive property.

$$\frac{8x^{1}\times 5x^{0}-\left(5x^{1}\times 8x^{0}+5\times 8x^{0}\right)}{\left(8x^{1}\right)^{2}}$$

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

$$\frac{8\times 5x^{1}-\left(5\times 8x^{1}+5\times 8x^{0}\right)}{\left(8x^{1}\right)^{2}}$$

Do the arithmetic.

$$\frac{40x^{1}-\left(40x^{1}+40x^{0}\right)}{\left(8x^{1}\right)^{2}}$$

Remove unnecessary parentheses.

$$\frac{40x^{1}-40x^{1}-40x^{0}}{\left(8x^{1}\right)^{2}}$$

Combine like terms.

$$\frac{\left(40-40\right)x^{1}-40x^{0}}{\left(8x^{1}\right)^{2}}$$

Subtract 40 from 40.

$$-\frac{40x^{0}}{\left(8x^{1}\right)^{2}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$-\frac{40x^{0}}{8^{2}x^{2}}$$

Raise 8 to the power 2.

$$-\frac{40x^{0}}{64x^{2}}$$

Multiply 1 times 2.

$$\frac{-40x^{0}}{64x^{2}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\left(-\frac{40}{64}\right)x^{-2}$$

Do the arithmetic.

$$-\frac{5}{8}x^{-2}$$