Calculus Examples

$g (y) = e^{y} \cdot y^{e}$

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 $f (y) = e^{y}$ and $g (y) = y^{e}$ .

$e^{y} \frac{d}{d y} [y^{e}] + y^{e} \frac{d}{d y} [e^{y}]$

Step 2

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

$e^{y} (e y^{e - 1}) + y^{e} \frac{d}{d y} [e^{y}]$

Step 3

Multiply $e^{y}$ by $e$ by adding the exponents.

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Step 3.1

Move $e$ .

$e e^{y} y^{e - 1} + y^{e} \frac{d}{d y} [e^{y}]$

Step 3.2

Multiply $e$ by $e^{y}$ .

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Step 3.2.1

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

$e^{1} e^{y} y^{e - 1} + y^{e} \frac{d}{d y} [e^{y}]$

Step 3.2.2

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

$e^{1 + y} y^{e - 1} + y^{e} \frac{d}{d y} [e^{y}]$

Step 4

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

$e^{1 + y} y^{e - 1} + y^{e} e^{y}$

Step 5

Reorder terms.

$e^{1 + y} y^{e - 1} + e^{y} y^{e}$