Calculus Examples

$y^{\frac{3}{2}} (y + c e^{y})$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y^{\frac{3}{2}}$ and $g (y) = y + c e^{y}$ .

$y^{\frac{3}{2}} \frac{d}{d y} [y + c e^{y}] + (y + c e^{y}) \frac{d}{d y} [y^{\frac{3}{2}}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $y + c e^{y}$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [c e^{y}]$ .

$y^{\frac{3}{2}} (\frac{d}{d y} [y] + \frac{d}{d y} [c e^{y}]) + (y + c e^{y}) \frac{d}{d y} [y^{\frac{3}{2}}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$y^{\frac{3}{2}} (1 + \frac{d}{d y} [c e^{y}]) + (y + c e^{y}) \frac{d}{d y} [y^{\frac{3}{2}}]$

Step 2.3

Since $c$ is constant with respect to $y$ , the derivative of $c e^{y}$ with respect to $y$ is $c \frac{d}{d y} [e^{y}]$ .

$y^{\frac{3}{2}} (1 + c \frac{d}{d y} [e^{y}]) + (y + c e^{y}) \frac{d}{d y} [y^{\frac{3}{2}}]$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d y} [a^{y}]$ is $a^{y} \ln (a)$ where $a$ = $e$ .

$y^{\frac{3}{2}} (1 + c e^{y}) + (y + c e^{y}) \frac{d}{d y} [y^{\frac{3}{2}}]$

Step 4

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = \frac{3}{2}$ .

$y^{\frac{3}{2}} (1 + c e^{y}) + (y + c e^{y}) (\frac{3}{2} y^{\frac{3}{2} - 1})$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y^{\frac{3}{2}} (1 + c e^{y}) + (y + c e^{y}) (\frac{3}{2} y^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 6

Combine $- 1$ and $\frac{2}{2}$ .

$y^{\frac{3}{2}} (1 + c e^{y}) + (y + c e^{y}) (\frac{3}{2} y^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 7

Combine the numerators over the common denominator.

$y^{\frac{3}{2}} (1 + c e^{y}) + (y + c e^{y}) (\frac{3}{2} y^{\frac{3 - 1 \cdot 2}{2}})$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$y^{\frac{3}{2}} (1 + c e^{y}) + (y + c e^{y}) (\frac{3}{2} y^{\frac{3 - 2}{2}})$

Step 8.2

Subtract $2$ from $3$ .

$y^{\frac{3}{2}} (1 + c e^{y}) + (y + c e^{y}) (\frac{3}{2} y^{\frac{1}{2}})$

Step 9

Combine $\frac{3}{2}$ and $y^{\frac{1}{2}}$ .

$y^{\frac{3}{2}} (1 + c e^{y}) + (y + c e^{y}) \frac{3 y^{\frac{1}{2}}}{2}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$y^{\frac{3}{2}} \cdot 1 + y^{\frac{3}{2}} (c e^{y}) + (y + c e^{y}) \frac{3 y^{\frac{1}{2}}}{2}$

Step 10.2

Apply the distributive property.

$y^{\frac{3}{2}} \cdot 1 + y^{\frac{3}{2}} (c e^{y}) + y \frac{3 y^{\frac{1}{2}}}{2} + c e^{y} \frac{3 y^{\frac{1}{2}}}{2}$

Step 10.3

Combine terms.

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Step 10.3.1

Multiply $y^{\frac{3}{2}}$ by $1$ .

$y^{\frac{3}{2}} + y^{\frac{3}{2}} (c e^{y}) + y \frac{3 y^{\frac{1}{2}}}{2} + c e^{y} \frac{3 y^{\frac{1}{2}}}{2}$

Step 10.3.2

Combine $y$ and $\frac{3 y^{\frac{1}{2}}}{2}$ .

$y^{\frac{3}{2}} + y^{\frac{3}{2}} (c e^{y}) + \frac{y (3 y^{\frac{1}{2}})}{2} + c e^{y} \frac{3 y^{\frac{1}{2}}}{2}$

Step 10.3.3

Raise $y$ to the power of $1$ .

$y^{\frac{3}{2}} + y^{\frac{3}{2}} (c e^{y}) + \frac{3 (y^{1} y^{\frac{1}{2}})}{2} + c e^{y} \frac{3 y^{\frac{1}{2}}}{2}$

Step 10.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{\frac{3}{2}} + y^{\frac{3}{2}} (c e^{y}) + \frac{3 y^{1 + \frac{1}{2}}}{2} + c e^{y} \frac{3 y^{\frac{1}{2}}}{2}$

Step 10.3.5

Write $1$ as a fraction with a common denominator.

$y^{\frac{3}{2}} + y^{\frac{3}{2}} (c e^{y}) + \frac{3 y^{\frac{2}{2} + \frac{1}{2}}}{2} + c e^{y} \frac{3 y^{\frac{1}{2}}}{2}$

Step 10.3.6

Combine the numerators over the common denominator.

$y^{\frac{3}{2}} + y^{\frac{3}{2}} (c e^{y}) + \frac{3 y^{\frac{2 + 1}{2}}}{2} + c e^{y} \frac{3 y^{\frac{1}{2}}}{2}$

Step 10.3.7

Add $2$ and $1$ .

$y^{\frac{3}{2}} + y^{\frac{3}{2}} (c e^{y}) + \frac{3 y^{\frac{3}{2}}}{2} + c e^{y} \frac{3 y^{\frac{1}{2}}}{2}$

Step 10.3.8

Combine $\frac{3 y^{\frac{1}{2}}}{2}$ and $c$ .

$y^{\frac{3}{2}} + y^{\frac{3}{2}} (c e^{y}) + \frac{3 y^{\frac{3}{2}}}{2} + \frac{3 y^{\frac{1}{2}} c}{2} e^{y}$

Step 10.3.9

Combine $\frac{3 y^{\frac{1}{2}} c}{2}$ and $e^{y}$ .

$y^{\frac{3}{2}} + y^{\frac{3}{2}} (c e^{y}) + \frac{3 y^{\frac{3}{2}}}{2} + \frac{3 y^{\frac{1}{2}} c e^{y}}{2}$

Step 10.3.10

To write $y^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y^{\frac{3}{2}} (c e^{y}) + y^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{3 y^{\frac{3}{2}}}{2} + \frac{3 y^{\frac{1}{2}} c e^{y}}{2}$

Step 10.3.11

Combine $y^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$y^{\frac{3}{2}} (c e^{y}) + \frac{y^{\frac{3}{2}} \cdot 2}{2} + \frac{3 y^{\frac{3}{2}}}{2} + \frac{3 y^{\frac{1}{2}} c e^{y}}{2}$

Step 10.3.12

Combine the numerators over the common denominator.

$y^{\frac{3}{2}} (c e^{y}) + \frac{y^{\frac{3}{2}} \cdot 2 + 3 y^{\frac{3}{2}}}{2} + \frac{3 y^{\frac{1}{2}} c e^{y}}{2}$

Step 10.3.13

Move $2$ to the left of $y^{\frac{3}{2}}$ .

$y^{\frac{3}{2}} (c e^{y}) + \frac{2 \cdot y^{\frac{3}{2}} + 3 y^{\frac{3}{2}}}{2} + \frac{3 y^{\frac{1}{2}} c e^{y}}{2}$

Step 10.3.14

Add $2 y^{\frac{3}{2}}$ and $3 y^{\frac{3}{2}}$ .

$y^{\frac{3}{2}} (c e^{y}) + \frac{5 y^{\frac{3}{2}}}{2} + \frac{3 y^{\frac{1}{2}} c e^{y}}{2}$

Step 10.4

Reorder terms.

$e^{y} y^{\frac{3}{2}} c + \frac{3 y^{\frac{1}{2}} c e^{y}}{2} + \frac{5 y^{\frac{3}{2}}}{2}$