Solve 1/8x-6*4x/x+7

$$\frac{ 1 }{ 8x-6 } \times \frac{ 4x }{ x+7 }$$

Evaluate

$$\frac{2x}{\left(4x-3\right)\left(x+7\right)}$$

Short Solution Steps

Multiply \frac{1}{8x-6} times \frac{4x}{x+7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4x}{\left(8x-6\right)\left(x+7\right)}$$

Factor the expressions that are not already factored.

$$\frac{4x}{2\left(4x-3\right)\left(x+7\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2x}{\left(4x-3\right)\left(x+7\right)}$$

Expand the expression.

$$\frac{2x}{4x^{2}+25x-21}$$

Differentiate w.r.t. x

$$-\frac{2\left(4x^{2}+21\right)}{\left(\left(4x-3\right)\left(x+7\right)\right)^{2}}$$

Steps Using Derivative Rule for Quotient

Multiply \frac{1}{8x-6} times \frac{4x}{x+7} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x}{\left(8x-6\right)\left(x+7\right)})$$

Factor the expressions that are not already factored in \frac{4x}{\left(8x-6\right)\left(x+7\right)}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x}{2\left(4x-3\right)\left(x+7\right)})$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x}{\left(4x-3\right)\left(x+7\right)})$$

Apply the distributive property by multiplying each term of 4x-3 by each term of x+7.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x}{4x^{2}+28x-3x-21})$$

Combine 28x and -3x to get 25x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x}{4x^{2}+25x-21})$$

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.

$$\frac{\left(4x^{2}+25x^{1}-21\right)\frac{\mathrm{d}}{\mathrm{d}x}(2x^{1})-2x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(4x^{2}+25x^{1}-21)}{\left(4x^{2}+25x^{1}-21\right)^{2}}$$

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}.

$$\frac{\left(4x^{2}+25x^{1}-21\right)\times 2x^{1-1}-2x^{1}\left(2\times 4x^{2-1}+25x^{1-1}\right)}{\left(4x^{2}+25x^{1}-21\right)^{2}}$$

Simplify.

$$\frac{\left(4x^{2}+25x^{1}-21\right)\times 2x^{0}-2x^{1}\left(8x^{1}+25x^{0}\right)}{\left(4x^{2}+25x^{1}-21\right)^{2}}$$

Multiply 4x^{2}+25x^{1}-21 times 2x^{0}.

$$\frac{4x^{2}\times 2x^{0}+25x^{1}\times 2x^{0}-21\times 2x^{0}-2x^{1}\left(8x^{1}+25x^{0}\right)}{\left(4x^{2}+25x^{1}-21\right)^{2}}$$

Multiply 2x^{1} times 8x^{1}+25x^{0}.

$$\frac{4x^{2}\times 2x^{0}+25x^{1}\times 2x^{0}-21\times 2x^{0}-\left(2x^{1}\times 8x^{1}+2x^{1}\times 25x^{0}\right)}{\left(4x^{2}+25x^{1}-21\right)^{2}}$$

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

$$\frac{4\times 2x^{2}+25\times 2x^{1}-21\times 2x^{0}-\left(2\times 8x^{1+1}+2\times 25x^{1}\right)}{\left(4x^{2}+25x^{1}-21\right)^{2}}$$

Simplify.

$$\frac{8x^{2}+50x^{1}-42x^{0}-\left(16x^{2}+50x^{1}\right)}{\left(4x^{2}+25x^{1}-21\right)^{2}}$$

Combine like terms.

$$\frac{-8x^{2}-42x^{0}}{\left(4x^{2}+25x^{1}-21\right)^{2}}$$

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

$$\frac{-8x^{2}-42x^{0}}{\left(4x^{2}+25x-21\right)^{2}}$$

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

$$\frac{-8x^{2}-42}{\left(4x^{2}+25x-21\right)^{2}}$$