Solve (1/x+1)(x/-x-2)

$$( \frac { 1 } { x + 1 } ) ( \frac { x } { - x - 2 } )$$

Evaluate

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

Short Solution Steps

Multiply \frac{1}{x+1} times \frac{x}{-x-2} by multiplying numerator times numerator and denominator times denominator.

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

Apply the distributive property by multiplying each term of x+1 by each term of -x-2.

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

Multiply x and x to get x^{2}.

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

Combine -2x and -x to get -3x.

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

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Quotient

Multiply \frac{1}{x+1} times \frac{x}{-x-2} by multiplying numerator times numerator and denominator times denominator.

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

Apply the distributive property by multiplying each term of x+1 by each term of -x-2.

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

Multiply x and x to get x^{2}.

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

Combine -2x and -x to get -3x.

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

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.

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

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}.

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

Simplify.

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

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

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

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

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

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

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

Simplify.

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

Combine like terms.

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

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

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

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

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