Calculus Examples

$e^{13 x y}$

Step 1

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

$g (y) = 13 x y$ .

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Step 1.1

To apply the Chain Rule, set

$u$ as

$13 x y$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d y} [13 x y]$

Step 1.2

Differentiate using the Exponential Rule which states that

$\frac{d}{d u} [a^{u}]$ is

$a^{u} \ln (a)$ where

$a$ =

$e$ .

$e^{u} \frac{d}{d y} [13 x y]$

Step 1.3

Replace all occurrences of

$u$ with

$13 x y$ .

$e^{13 x y} \frac{d}{d y} [13 x y]$

Step 2

Differentiate.

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Step 2.1

Since

$13 x$ is constant with respect to

$y$ , the derivative of

$13 x y$ with respect to

$y$ is

$13 x \frac{d}{d y} [y]$ .

$e^{13 x y} (13 x \frac{d}{d y} [y])$

Step 2.2

Differentiate using the Power Rule which states that

$\frac{d}{d y} [y^{n}]$ is

$n y^{n - 1}$ where

$n = 1$ .

$e^{13 x y} (13 x \cdot 1)$

Step 2.3

Multiply

$13$ by

$1$ .

$e^{13 x y} (13 x)$

Step 3

Simplify.

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Step 3.1

Reorder the factors of

$e^{13 x y} (13 x)$ .

$13 e^{13 x y} x$

Step 3.2

Reorder factors in

$13 e^{13 x y} x$ .

$13 x e^{13 x y}$

$e^{13 x y}$

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