Calculus Examples

$e^{x y}$

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 $g (y) = x y$ .

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $x$ by $1$ .

$e^{x y} x$

Step 2.3.2

Reorder factors in $e^{x y} x$ .

$x e^{x y}$