Solve 1/x-2a+1/x+2a+8a^2/4a^2x-x^3

$$\frac { 1 } { x - 2 a } + \frac { 1 } { x + 2 a } + \frac { 8 a ^ { 2 } } { 4 a ^ { 2 } x - x ^ { 3 } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2a and x+2a is \left(x-2a\right)\left(x+2a\right). Multiply \frac{1}{x-2a} times \frac{x+2a}{x+2a}. Multiply \frac{1}{x+2a} times \frac{x-2a}{x-2a}.

$$\frac{x+2a}{\left(x-2a\right)\left(x+2a\right)}+\frac{x-2a}{\left(x-2a\right)\left(x+2a\right)}+\frac{8a^{2}}{4a^{2}x-x^{3}}$$

Since \frac{x+2a}{\left(x-2a\right)\left(x+2a\right)} and \frac{x-2a}{\left(x-2a\right)\left(x+2a\right)} have the same denominator, add them by adding their numerators.

$$\frac{x+2a+x-2a}{\left(x-2a\right)\left(x+2a\right)}+\frac{8a^{2}}{4a^{2}x-x^{3}}$$

Combine like terms in x+2a+x-2a.

$$\frac{2x}{\left(x-2a\right)\left(x+2a\right)}+\frac{8a^{2}}{4a^{2}x-x^{3}}$$

Factor 4a^{2}x-x^{3}.

$$\frac{2x}{\left(x-2a\right)\left(x+2a\right)}+\frac{8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2a\right)\left(x+2a\right) and x\left(x+2a\right)\left(-x+2a\right) is x\left(x+2a\right)\left(-x+2a\right). Multiply \frac{2x}{\left(x-2a\right)\left(x+2a\right)} times \frac{-x}{-x}.

$$\frac{2x\left(-1\right)x}{x\left(x+2a\right)\left(-x+2a\right)}+\frac{8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)}$$

Since \frac{2x\left(-1\right)x}{x\left(x+2a\right)\left(-x+2a\right)} and \frac{8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x\left(-1\right)x+8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)}$$

Do the multiplications in 2x\left(-1\right)x+8a^{2}.

$$\frac{-2x^{2}+8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)}$$

Factor the expressions that are not already factored in \frac{-2x^{2}+8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)}.

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

Extract the negative sign in -x-2a. Extract the negative sign in x-2a.

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

Cancel out \left(x+2a\right)\left(-x+2a\right) in both numerator and denominator.

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

Multiply -1 and -1 to get 1.

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

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2a and x+2a is \left(x-2a\right)\left(x+2a\right). Multiply \frac{1}{x-2a} times \frac{x+2a}{x+2a}. Multiply \frac{1}{x+2a} times \frac{x-2a}{x-2a}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x+2a}{\left(x-2a\right)\left(x+2a\right)}+\frac{x-2a}{\left(x-2a\right)\left(x+2a\right)}+\frac{8a^{2}}{4a^{2}x-x^{3}})$$

Since \frac{x+2a}{\left(x-2a\right)\left(x+2a\right)} and \frac{x-2a}{\left(x-2a\right)\left(x+2a\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x+2a+x-2a}{\left(x-2a\right)\left(x+2a\right)}+\frac{8a^{2}}{4a^{2}x-x^{3}})$$

Combine like terms in x+2a+x-2a.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x}{\left(x-2a\right)\left(x+2a\right)}+\frac{8a^{2}}{4a^{2}x-x^{3}})$$

Factor 4a^{2}x-x^{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x}{\left(x-2a\right)\left(x+2a\right)}+\frac{8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)})$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2a\right)\left(x+2a\right) and x\left(x+2a\right)\left(-x+2a\right) is x\left(x+2a\right)\left(-x+2a\right). Multiply \frac{2x}{\left(x-2a\right)\left(x+2a\right)} times \frac{-x}{-x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x\left(-1\right)x}{x\left(x+2a\right)\left(-x+2a\right)}+\frac{8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)})$$

Since \frac{2x\left(-1\right)x}{x\left(x+2a\right)\left(-x+2a\right)} and \frac{8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)} have the same denominator, add them by adding their numerators.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x\left(-1\right)x+8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)})$$

Do the multiplications in 2x\left(-1\right)x+8a^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{-2x^{2}+8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)})$$

Factor the expressions that are not already factored in \frac{-2x^{2}+8a^{2}}{x\left(x+2a\right)\left(-x+2a\right)}.

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

Extract the negative sign in -x-2a. Extract the negative sign in x-2a.

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

Cancel out \left(x+2a\right)\left(-x+2a\right) in both numerator and denominator.

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

Multiply -1 and -1 to get 1.

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

The derivative of ax^{n} is nax^{n-1}.

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

Subtract 1 from -1.

$$-2x^{-2}$$