Solve -frac{1/2sqrt{2}}{-1/2x-1/2sqrt{2}}

$$\frac { - \frac { 1 } { 2 } \sqrt { 2 } } { - \frac { 1 } { 2 } x - \frac { 1 } { 2 } \sqrt { 2 } }$$

Evaluate

$$\frac{\sqrt{2}}{x+\sqrt{2}}$$

Short Solution Steps

Rationalize the denominator of \frac{-\frac{1}{2}\sqrt{2}}{-\frac{1}{2}x-\frac{1}{2}\sqrt{2}} by multiplying numerator and denominator by -\frac{1}{2}x+\frac{1}{2}\sqrt{2}.

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

Consider \left(-\frac{1}{2}x-\frac{1}{2}\sqrt{2}\right)\left(-\frac{1}{2}x+\frac{1}{2}\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(-\frac{1}{2}x\right)^{2}.

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

Calculate -\frac{1}{2} to the power of 2 and get \frac{1}{4}.

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

Expand \left(-\frac{1}{2}\sqrt{2}\right)^{2}.

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

Calculate -\frac{1}{2} to the power of 2 and get \frac{1}{4}.

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

The square of \sqrt{2} is 2.

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

Multiply \frac{1}{4} and 2 to get \frac{2}{4}.

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

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

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

Use the distributive property to multiply -\frac{1}{2}\sqrt{2} by -\frac{1}{2}x+\frac{1}{2}\sqrt{2}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Multiply -\frac{1}{2} times -\frac{1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{-\left(-1\right)}{2\times 2}.

$$\frac{\frac{1}{4}\sqrt{2}x-\frac{1}{2}\times 2\times \frac{1}{2}}{\frac{1}{4}x^{2}-\frac{1}{2}}$$

Cancel out 2 and 2.

$$\frac{\frac{1}{4}\sqrt{2}x-\frac{1}{2}}{\frac{1}{4}x^{2}-\frac{1}{2}}$$

Factor the expressions that are not already factored.

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

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{\sqrt{2}x-2}{\left(\frac{1}{4}\right)^{0}\left(x^{2}-2\right)}$$

Expand the expression.

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

Differentiate w.r.t. x

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