Solve 2x/x^2-16y^2-1/x-4y

$$\frac{ 2x }{ { x }^{ 2 } -16 { y }^{ 2 } } - \frac{ 1 }{ x-4y }$$

Evaluate

$$\frac{1}{x+4y}$$

Short Solution Steps

Factor x^{2}-16y^{2}.

$$\frac{2x}{\left(x-4y\right)\left(x+4y\right)}-\frac{1}{x-4y}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4y\right)\left(x+4y\right) and x-4y is \left(x-4y\right)\left(x+4y\right). Multiply \frac{1}{x-4y} times \frac{x+4y}{x+4y}.

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

Since \frac{2x}{\left(x-4y\right)\left(x+4y\right)} and \frac{x+4y}{\left(x-4y\right)\left(x+4y\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2x-\left(x+4y\right).

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

Combine like terms in 2x-x-4y.

$$\frac{x-4y}{\left(x-4y\right)\left(x+4y\right)}$$

Cancel out x-4y in both numerator and denominator.

$$\frac{1}{x+4y}$$

Differentiate w.r.t. x

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