Calculus Examples

$3 e^{x - 3}$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 e^{x - 3}$ with respect to $x$ is $3 \frac{d}{d x} [e^{x - 3}]$ .

$3 \frac{d}{d x} [e^{x - 3}]$

Step 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 $g (x) = x - 3$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x - 3$ .

$3 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [x - 3])$

Step 2.2

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

$3 (e^{u} \frac{d}{d x} [x - 3])$

Step 2.3

Replace all occurrences of $u$ with $x - 3$ .

$3 (e^{x - 3} \frac{d}{d x} [x - 3])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$3 e^{x - 3} (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 3.2

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

$3 e^{x - 3} (1 + \frac{d}{d x} [- 3])$

Step 3.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$3 e^{x - 3} (1 + 0)$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

$3 e^{x - 3} \cdot 1$

Step 3.4.2

Multiply $3$ by $1$ .

$3 e^{x - 3}$