Calculus Examples

$y = e^{x + 1} + 1$

Step 1

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

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

Step 2

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

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = x + 1$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $x + 1$ .

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

Step 2.1.2

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

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

Step 2.1.3

Replace all occurrences of $u$ with $x + 1$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$e^{x + 1} (1 + 0) + \frac{d}{d x} [1]$

Step 2.5

Add $1$ and $0$ .

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

Step 2.6

Multiply $e^{x + 1}$ by $1$ .

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

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 $0$ .

$e^{x + 1} + 0$

Step 3.2

Add $e^{x + 1}$ and $0$ .

$e^{x + 1}$