Calculus Examples

$3 e^{3 x y} y$

Step 1

Since $3$ is constant with respect to $y$ , the derivative of $3 e^{3 x y} y$ with respect to $y$ is $3 \frac{d}{d y} [e^{3 x y} y]$ .

$3 \frac{d}{d y} [e^{3 x y} y]$

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 $f (y) = e^{3 x y}$ and $g (y) = y$ .

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

Step 3

Differentiate using the Power Rule.

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

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

Multiply $e^{3 x y}$ by $1$ .

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

Step 4

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 $g (y) = 3 x y$ .

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To apply the Chain Rule, set $u$ as $3 x y$ .

$3 (e^{3 x y} + y (\frac{d}{d u} [e^{u}] \frac{d}{d y} [3 x y]))$

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

$3 (e^{3 x y} + y (e^{u} \frac{d}{d y} [3 x y]))$

Replace all occurrences of $u$ with $3 x y$ .

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

Step 5

Differentiate.

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Since $3 x$ is constant with respect to $y$ , the derivative of $3 x y$ with respect to $y$ is $3 x \frac{d}{d y} [y]$ .

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

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

$3 (e^{3 x y} + y (e^{3 x y} (3 x \cdot 1)))$

Multiply $3$ by $1$ .

$3 (e^{3 x y} + y (e^{3 x y} (3 x)))$

Step 6

Simplify.

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Apply the distributive property.

$3 e^{3 x y} + 3 (y (e^{3 x y} (3 x)))$

Multiply $3$ by $3$ .

$3 e^{3 x y} + 9 y (e^{3 x y} (x))$

Reorder terms.

$9 e^{3 x y} x y + 3 e^{3 x y}$

Reorder factors in $9 e^{3 x y} x y + 3 e^{3 x y}$ .

$9 x y e^{3 x y} + 3 e^{3 x y}$