Calculus Examples

$2 e^{x} + 1$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Differentiate using the Constant Rule.

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Step 3.1

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

$2 e^{x} + 0$

Step 3.2

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

$2 e^{x}$