Solve 4/x^2+3xdiv8/x^2+5x+6

$$\frac { 4 } { x ^ { 2 } + 3 x } \div \frac { 8 } { x ^ { 2 } + 5 x + 6 }$$

Evaluate

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

Short Solution Steps

Divide \frac{4}{x^{2}+3x} by \frac{8}{x^{2}+5x+6} by multiplying \frac{4}{x^{2}+3x} by the reciprocal of \frac{8}{x^{2}+5x+6}.

$$\frac{4\left(x^{2}+5x+6\right)}{\left(x^{2}+3x\right)\times 8}$$

Cancel out 4 in both numerator and denominator.

$$\frac{x^{2}+5x+6}{2\left(x^{2}+3x\right)}$$

Factor the expressions that are not already factored.

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

Cancel out x+3 in both numerator and denominator.

$$\frac{x+2}{2x}$$

Differentiate w.r.t. x

$$-\frac{1}{x^{2}}$$

Steps Using Derivative Rule for Quotient

Divide \frac{4}{x^{2}+3x} by \frac{8}{x^{2}+5x+6} by multiplying \frac{4}{x^{2}+3x} by the reciprocal of \frac{8}{x^{2}+5x+6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4\left(x^{2}+5x+6\right)}{\left(x^{2}+3x\right)\times 8})$$

Cancel out 4 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}+5x+6}{2\left(x^{2}+3x\right)})$$

Factor the expressions that are not already factored in \frac{x^{2}+5x+6}{2\left(x^{2}+3x\right)}.

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

Cancel out x+3 in both numerator and denominator.

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

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.

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

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}.

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

Do the arithmetic.

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

Expand using distributive property.

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

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

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

Do the arithmetic.

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

Remove unnecessary parentheses.

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

Combine like terms.

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

Subtract 2 from 2.

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

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

$$-\frac{4x^{0}}{2^{2}x^{2}}$$

Raise 2 to the power 2.

$$-\frac{4x^{0}}{4x^{2}}$$

Multiply 1 times 2.

$$\frac{-4x^{0}}{4x^{2}}$$

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

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

Do the arithmetic.

$$-x^{-2}$$