Solve 4/x+4/frac{1{x+4}-1/x}

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

Differentiate w.r.t. x

$$-1$$

Steps Using Definition of a Derivative

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

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

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

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

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

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

Combine like terms in x-x-4.

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

Divide \frac{4}{x+4} by \frac{-4}{x\left(x+4\right)} by multiplying \frac{4}{x+4} by the reciprocal of \frac{-4}{x\left(x+4\right)}.

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

Cancel out 4\left(x+4\right) in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x}{-1})$$

Anything divided by -1 gives its opposite.

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

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

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

Subtract 1 from 1.

$$-x^{0}$$

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

$$-1$$

Evaluate

$$-x$$