Solve x-2/x^2-9*x^2-4/(2x-4)divx-2/x-3

$$\frac { x - 2 } { x ^ { 2 } - 9 } \cdot \frac { x ^ { 2 - 4 } } { ( 2 x - 4 ) } \div \frac { x - 2 } { x - 3 }$$

Evaluate

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

Short Solution Steps

Subtract 4 from 2 to get -2.

$$\frac{\frac{x-2}{x^{2}-9}\times \frac{x^{-2}}{2x-4}}{\frac{x-2}{x-3}}$$

Multiply \frac{x-2}{x^{2}-9} times \frac{x^{-2}}{2x-4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x-2\right)x^{-2}}{\left(x^{2}-9\right)\left(2x-4\right)}}{\frac{x-2}{x-3}}$$

Divide \frac{\left(x-2\right)x^{-2}}{\left(x^{2}-9\right)\left(2x-4\right)} by \frac{x-2}{x-3} by multiplying \frac{\left(x-2\right)x^{-2}}{\left(x^{2}-9\right)\left(2x-4\right)} by the reciprocal of \frac{x-2}{x-3}.

$$\frac{\left(x-2\right)x^{-2}\left(x-3\right)}{\left(x^{2}-9\right)\left(2x-4\right)\left(x-2\right)}$$

Cancel out x-2 in both numerator and denominator.

$$\frac{x^{-2}\left(x-3\right)}{\left(2x-4\right)\left(x^{2}-9\right)}$$

Factor the expressions that are not already factored.

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

Cancel out x-3 in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{-2}}{2x^{2}+2x-12}$$

Differentiate w.r.t. x

$$-\frac{\left(8x+3\right)^{2}-201}{32x^{3}\left(\left(x-2\right)\left(x+3\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Subtract 4 from 2 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{x-2}{x^{2}-9}\times \frac{x^{-2}}{2x-4}}{\frac{x-2}{x-3}})$$

Multiply \frac{x-2}{x^{2}-9} times \frac{x^{-2}}{2x-4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{\left(x-2\right)x^{-2}}{\left(x^{2}-9\right)\left(2x-4\right)}}{\frac{x-2}{x-3}})$$

Divide \frac{\left(x-2\right)x^{-2}}{\left(x^{2}-9\right)\left(2x-4\right)} by \frac{x-2}{x-3} by multiplying \frac{\left(x-2\right)x^{-2}}{\left(x^{2}-9\right)\left(2x-4\right)} by the reciprocal of \frac{x-2}{x-3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-2\right)x^{-2}\left(x-3\right)}{\left(x^{2}-9\right)\left(2x-4\right)\left(x-2\right)})$$

Cancel out x-2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-2}\left(x-3\right)}{\left(2x-4\right)\left(x^{2}-9\right)})$$

Factor the expressions that are not already factored in \frac{x^{-2}\left(x-3\right)}{\left(2x-4\right)\left(x^{2}-9\right)}.

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

Cancel out x-3 in both numerator and denominator.

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

Use the distributive property to multiply 2 by x-2.

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

Use the distributive property to multiply 2x-4 by x+3 and combine like terms.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{-2}}{2x^{2}+2x-12})$$

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.

$$\frac{\left(2x^{2}+2x^{1}-12\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{-2})-x^{-2}\frac{\mathrm{d}}{\mathrm{d}x}(2x^{2}+2x^{1}-12)}{\left(2x^{2}+2x^{1}-12\right)^{2}}$$

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}.

$$\frac{\left(2x^{2}+2x^{1}-12\right)\left(-2\right)x^{-2-1}-x^{-2}\left(2\times 2x^{2-1}+2x^{1-1}\right)}{\left(2x^{2}+2x^{1}-12\right)^{2}}$$

Simplify.

$$\frac{\left(2x^{2}+2x^{1}-12\right)\left(-2\right)x^{-3}-x^{-2}\left(4x^{1}+2x^{0}\right)}{\left(2x^{2}+2x^{1}-12\right)^{2}}$$

Multiply 2x^{2}+2x^{1}-12 times -2x^{-3}.

$$\frac{2x^{2}\left(-2\right)x^{-3}+2x^{1}\left(-2\right)x^{-3}-12\left(-2\right)x^{-3}-x^{-2}\left(4x^{1}+2x^{0}\right)}{\left(2x^{2}+2x^{1}-12\right)^{2}}$$

Multiply x^{-2} times 4x^{1}+2x^{0}.

$$\frac{2x^{2}\left(-2\right)x^{-3}+2x^{1}\left(-2\right)x^{-3}-12\left(-2\right)x^{-3}-\left(x^{-2}\times 4x^{1}+x^{-2}\times 2x^{0}\right)}{\left(2x^{2}+2x^{1}-12\right)^{2}}$$

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

$$\frac{2\left(-2\right)x^{2-3}+2\left(-2\right)x^{1-3}-12\left(-2\right)x^{-3}-\left(4x^{-2+1}+2x^{-2}\right)}{\left(2x^{2}+2x^{1}-12\right)^{2}}$$

Simplify.

$$\frac{-4\times \frac{1}{x}-4x^{-2}+24x^{-3}-\left(4\times \frac{1}{x}+2x^{-2}\right)}{\left(2x^{2}+2x^{1}-12\right)^{2}}$$

Combine like terms.

$$\frac{-8\times \frac{1}{x}-6x^{-2}+24x^{-3}}{\left(2x^{2}+2x^{1}-12\right)^{2}}$$

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

$$\frac{-8\times \frac{1}{x}-6x^{-2}+24x^{-3}}{\left(2x^{2}+2x-12\right)^{2}}$$