Solve x^2/x^2-49div2x^3/x^2-16x+63

$$\frac { x ^ { 2 } } { x ^ { 2 } - 49 } \div \frac { 2 x ^ { 3 } } { x ^ { 2 } - 16 x + 63 }$$

Evaluate

$$\frac{x-9}{2x\left(x+7\right)}$$

Short Solution Steps

Divide \frac{x^{2}}{x^{2}-49} by \frac{2x^{3}}{x^{2}-16x+63} by multiplying \frac{x^{2}}{x^{2}-49} by the reciprocal of \frac{2x^{3}}{x^{2}-16x+63}.

$$\frac{x^{2}\left(x^{2}-16x+63\right)}{\left(x^{2}-49\right)\times 2x^{3}}$$

Cancel out x^{2} in both numerator and denominator.

$$\frac{x^{2}-16x+63}{2x\left(x^{2}-49\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(x-9\right)\left(x-7\right)}{2x\left(x-7\right)\left(x+7\right)}$$

Cancel out x-7 in both numerator and denominator.

$$\frac{x-9}{2x\left(x+7\right)}$$

Expand the expression.

$$\frac{x-9}{2x^{2}+14x}$$

Differentiate w.r.t. x

$$\frac{\left(-x-3\right)\left(x-21\right)}{2\left(x\left(x+7\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Divide \frac{x^{2}}{x^{2}-49} by \frac{2x^{3}}{x^{2}-16x+63} by multiplying \frac{x^{2}}{x^{2}-49} by the reciprocal of \frac{2x^{3}}{x^{2}-16x+63}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}\left(x^{2}-16x+63\right)}{\left(x^{2}-49\right)\times 2x^{3}})$$

Cancel out x^{2} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{2}-16x+63}{2x\left(x^{2}-49\right)})$$

Factor the expressions that are not already factored in \frac{x^{2}-16x+63}{2x\left(x^{2}-49\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(x-9\right)\left(x-7\right)}{2x\left(x-7\right)\left(x+7\right)})$$

Cancel out x-7 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x-9}{2x\left(x+7\right)})$$

Use the distributive property to multiply 2x by x+7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x-9}{2x^{2}+14x})$$

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.

$$\frac{\left(2x^{2}+14x^{1}\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}-9)-\left(x^{1}-9\right)\frac{\mathrm{d}}{\mathrm{d}x}(2x^{2}+14x^{1})}{\left(2x^{2}+14x^{1}\right)^{2}}$$

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}.

$$\frac{\left(2x^{2}+14x^{1}\right)x^{1-1}-\left(x^{1}-9\right)\left(2\times 2x^{2-1}+14x^{1-1}\right)}{\left(2x^{2}+14x^{1}\right)^{2}}$$

Simplify.

$$\frac{\left(2x^{2}+14x^{1}\right)x^{0}-\left(x^{1}-9\right)\left(4x^{1}+14x^{0}\right)}{\left(2x^{2}+14x^{1}\right)^{2}}$$

Multiply 2x^{2}+14x^{1} times x^{0}.

$$\frac{2x^{2}x^{0}+14x^{1}x^{0}-\left(x^{1}-9\right)\left(4x^{1}+14x^{0}\right)}{\left(2x^{2}+14x^{1}\right)^{2}}$$

Multiply x^{1}-9 times 4x^{1}+14x^{0}.

$$\frac{2x^{2}x^{0}+14x^{1}x^{0}-\left(x^{1}\times 4x^{1}+x^{1}\times 14x^{0}-9\times 4x^{1}-9\times 14x^{0}\right)}{\left(2x^{2}+14x^{1}\right)^{2}}$$

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

$$\frac{2x^{2}+14x^{1}-\left(4x^{1+1}+14x^{1}-9\times 4x^{1}-9\times 14x^{0}\right)}{\left(2x^{2}+14x^{1}\right)^{2}}$$

Simplify.

$$\frac{2x^{2}+14x^{1}-\left(4x^{2}+14x^{1}-36x^{1}-126x^{0}\right)}{\left(2x^{2}+14x^{1}\right)^{2}}$$

Combine like terms.

$$\frac{-2x^{2}+36x^{1}+126x^{0}}{\left(2x^{2}+14x^{1}\right)^{2}}$$

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

$$\frac{-2x^{2}+36x+126x^{0}}{\left(2x^{2}+14x\right)^{2}}$$

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

$$\frac{-2x^{2}+36x+126\times 1}{\left(2x^{2}+14x\right)^{2}}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{-2x^{2}+36x+126}{\left(2x^{2}+14x\right)^{2}}$$