Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $x e^{y} + y e^{x}$ with respect to $x$ is $\frac{d}{d x} [x e^{y}] + \frac{d}{d x} [y e^{x}]$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

Reorder terms.

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

Step 4.2

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

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