Calculus Examples

$1 + 2 x e^{y}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $1 + 2 x e^{y}$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [2 x e^{y}]$ .

$\frac{d}{d y} [1] + \frac{d}{d y} [2 x e^{y}]$

Step 1.2

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [2 x e^{y}]$

Step 2

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

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Step 2.1

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

$0 + 2 x \frac{d}{d y} [e^{y}]$

Step 2.2

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

$0 + 2 x e^{y}$

Step 3

Simplify.

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Step 3.1

Add $0$ and $2 x e^{y}$ .

$2 x e^{y}$

Step 3.2

Reorder the factors of $2 x e^{y}$ .

$2 e^{y} x$

Step 3.3

Reorder factors in $2 e^{y} x$ .

$2 x e^{y}$