Calculus Examples

$f (x) = (x - 1) e^{3 x + 2}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 1$ and $g (x) = e^{3 x + 2}$ .

$(x - 1) \frac{d}{d x} [e^{3 x + 2}] + e^{3 x + 2} \frac{d}{d x} [x - 1]$

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 $g (x) = 3 x + 2$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x + 2$ .

$(x - 1) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [3 x + 2]) + e^{3 x + 2} \frac{d}{d x} [x - 1]$

Step 2.2

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

$(x - 1) (e^{u} \frac{d}{d x} [3 x + 2]) + e^{3 x + 2} \frac{d}{d x} [x - 1]$

Step 2.3

Replace all occurrences of $u$ with $3 x + 2$ .

$(x - 1) (e^{3 x + 2} \frac{d}{d x} [3 x + 2]) + e^{3 x + 2} \frac{d}{d x} [x - 1]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$ .

$(x - 1) e^{3 x + 2} (\frac{d}{d x} [3 x] + \frac{d}{d x} [2]) + e^{3 x + 2} \frac{d}{d x} [x - 1]$

Step 3.2

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

$(x - 1) e^{3 x + 2} (3 \frac{d}{d x} [x] + \frac{d}{d x} [2]) + e^{3 x + 2} \frac{d}{d x} [x - 1]$

Step 3.3

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

$(x - 1) e^{3 x + 2} (3 \cdot 1 + \frac{d}{d x} [2]) + e^{3 x + 2} \frac{d}{d x} [x - 1]$

Step 3.4

Multiply $3$ by $1$ .

$(x - 1) e^{3 x + 2} (3 + \frac{d}{d x} [2]) + e^{3 x + 2} \frac{d}{d x} [x - 1]$

Step 3.5

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

$(x - 1) e^{3 x + 2} (3 + 0) + e^{3 x + 2} \frac{d}{d x} [x - 1]$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $3$ and $0$ .

$(x - 1) e^{3 x + 2} \cdot 3 + e^{3 x + 2} \frac{d}{d x} [x - 1]$

Step 3.6.2

Move $3$ to the left of $(x - 1) e^{3 x + 2}$ .

$3 \cdot ((x - 1) e^{3 x + 2}) + e^{3 x + 2} \frac{d}{d x} [x - 1]$

Step 3.7

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

$3 (x - 1) e^{3 x + 2} + e^{3 x + 2} (\frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 3.8

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

$3 (x - 1) e^{3 x + 2} + e^{3 x + 2} (1 + \frac{d}{d x} [- 1])$

Step 3.9

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

$3 (x - 1) e^{3 x + 2} + e^{3 x + 2} (1 + 0)$

Step 3.10

Simplify the expression.

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Step 3.10.1

Add $1$ and $0$ .

$3 (x - 1) e^{3 x + 2} + e^{3 x + 2} \cdot 1$

Step 3.10.2

Multiply $e^{3 x + 2}$ by $1$ .

$3 (x - 1) e^{3 x + 2} + e^{3 x + 2}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(3 x + 3 \cdot - 1) e^{3 x + 2} + e^{3 x + 2}$

Step 4.2

Multiply $3$ by $- 1$ .

$(3 x - 3) e^{3 x + 2} + e^{3 x + 2}$

Step 4.3

Reorder terms.

$e^{3 x + 2} (3 x - 3) + e^{3 x + 2}$

Step 4.4

Simplify each term.

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Step 4.4.1

Apply the distributive property.

$e^{3 x + 2} (3 x) + e^{3 x + 2} \cdot - 3 + e^{3 x + 2}$

Step 4.4.2

Rewrite using the commutative property of multiplication.

$3 e^{3 x + 2} x + e^{3 x + 2} \cdot - 3 + e^{3 x + 2}$

Step 4.4.3

Move $- 3$ to the left of $e^{3 x + 2}$ .

$3 e^{3 x + 2} x - 3 e^{3 x + 2} + e^{3 x + 2}$

Step 4.5

Add $- 3 e^{3 x + 2}$ and $e^{3 x + 2}$ .

$3 e^{3 x + 2} x - 2 e^{3 x + 2}$

Step 4.6

Reorder factors in $3 e^{3 x + 2} x - 2 e^{3 x + 2}$ .

$3 x e^{3 x + 2} - 2 e^{3 x + 2}$