Solve 2x/x^2-64y^2-1/x-8y

$$\frac { 2 x } { x ^ { 2 } - 64 y ^ { 2 } } - \frac { 1 } { x - 8 y }$$

Evaluate

$$\frac{1}{x+8y}$$

Short Solution Steps

Factor x^{2}-64y^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-8y\right)\left(x+8y\right) and x-8y is \left(x-8y\right)\left(x+8y\right). Multiply \frac{1}{x-8y} times \frac{x+8y}{x+8y}.

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

Since \frac{2x}{\left(x-8y\right)\left(x+8y\right)} and \frac{x+8y}{\left(x-8y\right)\left(x+8y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2x-\left(x+8y\right).

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

Combine like terms in 2x-x-8y.

$$\frac{x-8y}{\left(x-8y\right)\left(x+8y\right)}$$

Cancel out x-8y in both numerator and denominator.

$$\frac{1}{x+8y}$$

Differentiate w.r.t. x

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