Calculus Examples

$y = 7 e^{x + 2} + 2 x$

Step 1

By the Sum Rule, the derivative of $7 e^{x + 2} + 2 x$ with respect to $x$ is $\frac{d}{d x} [7 e^{x + 2}] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [7 e^{x + 2}] + \frac{d}{d x} [2 x]$

Step 2

Evaluate $\frac{d}{d x} [7 e^{x + 2}]$ .

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Step 2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 e^{x + 2}$ with respect to $x$ is $7 \frac{d}{d x} [e^{x + 2}]$ .

$7 \frac{d}{d x} [e^{x + 2}] + \frac{d}{d x} [2 x]$

Step 2.2

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 $g (x) = x + 2$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x + 2$ .

$7 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x + 2]) + \frac{d}{d x} [2 x]$

Step 2.2.2

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

$7 (e^{u} \frac{d}{d x} [x + 2]) + \frac{d}{d x} [2 x]$

Step 2.2.3

Replace all occurrences of $u$ with $x + 2$ .

$7 (e^{x + 2} \frac{d}{d x} [x + 2]) + \frac{d}{d x} [2 x]$

Step 2.3

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

$7 (e^{x + 2} (\frac{d}{d x} [x] + \frac{d}{d x} [2])) + \frac{d}{d x} [2 x]$

Step 2.4

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

$7 (e^{x + 2} (1 + \frac{d}{d x} [2])) + \frac{d}{d x} [2 x]$

Step 2.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$7 (e^{x + 2} (1 + 0)) + \frac{d}{d x} [2 x]$

Step 2.6

Add $1$ and $0$ .

$7 (e^{x + 2} \cdot 1) + \frac{d}{d x} [2 x]$

Step 2.7

Multiply $e^{x + 2}$ by $1$ .

$7 e^{x + 2} + \frac{d}{d x} [2 x]$

Step 3

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

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Step 3.1

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

$7 e^{x + 2} + 2 \frac{d}{d x} [x]$

Step 3.2

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

$7 e^{x + 2} + 2 \cdot 1$

Step 3.3

Multiply $2$ by $1$ .

$7 e^{x + 2} + 2$