Solve frac{sqrt{y}}{y}*frac{sqrt{y}}{y}

$$\frac { \sqrt { y } } { y } \times \frac { \sqrt { y } } { y }$$

Evaluate

$$\frac{1}{y}$$

Solution Steps

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

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

To raise \frac{\sqrt{y}}{y} to a power, raise both numerator and denominator to the power and then divide.

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

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

$$\frac{y}{y^{2}}$$

Cancel out y in both numerator and denominator.

$$\frac{1}{y}$$

Differentiate w.r.t. y

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

Steps Using Definition of a Derivative

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

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

To raise \frac{\sqrt{y}}{y} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Cancel out y in both numerator and denominator.

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

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

$$-y^{-1-1}$$

Subtract 1 from -1.

$$-y^{-2}$$