Calculus Examples

$2 y e^{y^{2}}$

Step 1

Since $2$ is constant with respect to $y$ , the derivative of $2 y e^{y^{2}}$ with respect to $y$ is $2 \frac{d}{d y} [y e^{y^{2}}]$ .

$2 \frac{d}{d y} [y e^{y^{2}}]$

Step 2

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 $f (y) = y$ and $g (y) = e^{y^{2}}$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = e^{y}$ and $g (y) = y^{2}$ .

Tap for more steps...

Step 3.1

To apply the Chain Rule, set $u$ as $y^{2}$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $y^{2}$ .

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the expression.

Tap for more steps...

Step 8.1

Add $1$ and $1$ .

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

Step 8.2

Move $2$ to the left of $e^{y^{2}}$ .

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

Step 9

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

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

Step 10

Multiply $e^{y^{2}}$ by $1$ .

$2 (y^{2} (2 e^{y^{2}}) + e^{y^{2}})$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Apply the distributive property.

$2 (y^{2} (2 e^{y^{2}})) + 2 e^{y^{2}}$

Step 11.2

Multiply $2$ by $2$ .

$4 y^{2} e^{y^{2}} + 2 e^{y^{2}}$

Step 11.3

Reorder terms.

$4 e^{y^{2}} y^{2} + 2 e^{y^{2}}$

Step 11.4

Reorder factors in $4 e^{y^{2}} y^{2} + 2 e^{y^{2}}$ .

$4 y^{2} e^{y^{2}} + 2 e^{y^{2}}$