Solve y^frac{5{2}}*y^-frac{1{2}}*y^frac{5{2}}

$$y ^ { \frac { 5 } { 2 } } \cdot y ^ { - \frac { 1 } { 2 } } \cdot y ^ { \frac { 5 } { 2 } }$$

Evaluate

$$y^{\frac{9}{2}}$$

Steps Using Product of Powers Property

Use the rules of exponents to simplify the expression.

$$y^{\frac{5}{2}}y^{-\frac{1}{2}}y^{\frac{5}{2}}$$

Use the Multiplication Rule for Exponents.

$$y^{\frac{5-1+5}{2}}$$

Add the exponents \frac{5}{2} and -\frac{1}{2}.

$$y^{2+\frac{5}{2}}$$

Add the exponents 2 and \frac{5}{2}.

$$y^{\frac{9}{2}}$$

Differentiate w.r.t. y

$$\frac{9y^{\frac{7}{2}}}{2}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{5}{2} and -\frac{1}{2} to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{2}y^{\frac{5}{2}})$$

To multiply powers of the same base, add their exponents. Add 2 and \frac{5}{2} to get \frac{9}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}y}(y^{\frac{9}{2}})$$

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

$$\frac{9}{2}y^{\frac{9}{2}-1}$$

Subtract 1 from \frac{9}{2}.

$$\frac{9}{2}y^{\frac{7}{2}}$$