Solve 1/sqrt{x^2}x/sqrt{x^2}

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

Evaluate

$$\frac{1}{x}$$

Solution Steps

Multiply \frac{1}{\sqrt{x^{2}}} times \frac{x}{\sqrt{x^{2}}} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \sqrt{x^{2}} and \sqrt{x^{2}} to get \left(\sqrt{x^{2}}\right)^{2}.

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

Calculate \sqrt{x^{2}} to the power of 2 and get x^{2}.

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

Cancel out x in both numerator and denominator.

$$\frac{1}{x}$$

Differentiate w.r.t. x

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

Steps Using Definition of a Derivative

Multiply \frac{1}{\sqrt{x^{2}}} times \frac{x}{\sqrt{x^{2}}} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \sqrt{x^{2}} and \sqrt{x^{2}} to get \left(\sqrt{x^{2}}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x}{\left(\sqrt{x^{2}}\right)^{2}})$$

Calculate \sqrt{x^{2}} to the power of 2 and get x^{2}.

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

Cancel out x in both numerator and denominator.

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

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

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

Subtract 1 from -1.

$$-x^{-2}$$