Calculus Examples

$g (x) = e^{x} \ln (3 x + 11)$

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 $f (x) = e^{x}$ and $g (x) = \ln (3 x + 11)$ .

$e^{x} \frac{d}{d x} [\ln (3 x + 11)] + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 3 x + 11$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x + 11$ .

$e^{x} (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [3 x + 11]) + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$e^{x} (\frac{1}{u} \frac{d}{d x} [3 x + 11]) + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 2.3

Replace all occurrences of $u$ with $3 x + 11$ .

$e^{x} (\frac{1}{3 x + 11} \frac{d}{d x} [3 x + 11]) + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{3 x + 11}$ and $e^{x}$ .

$\frac{e^{x}}{3 x + 11} \frac{d}{d x} [3 x + 11] + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 3.2

By the Sum Rule, the derivative of $3 x + 11$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [11]$ .

$\frac{e^{x}}{3 x + 11} (\frac{d}{d x} [3 x] + \frac{d}{d x} [11]) + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 3.3

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

$\frac{e^{x}}{3 x + 11} (3 \frac{d}{d x} [x] + \frac{d}{d x} [11]) + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 3.4

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

$\frac{e^{x}}{3 x + 11} (3 \cdot 1 + \frac{d}{d x} [11]) + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 3.5

Multiply $3$ by $1$ .

$\frac{e^{x}}{3 x + 11} (3 + \frac{d}{d x} [11]) + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 3.6

Since $11$ is constant with respect to $x$ , the derivative of $11$ with respect to $x$ is $0$ .

$\frac{e^{x}}{3 x + 11} (3 + 0) + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 3.7

Combine fractions.

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Step 3.7.1

Add $3$ and $0$ .

$\frac{e^{x}}{3 x + 11} \cdot 3 + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 3.7.2

Combine $\frac{e^{x}}{3 x + 11}$ and $3$ .

$\frac{e^{x} \cdot 3}{3 x + 11} + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 3.7.3

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

$\frac{3 \cdot e^{x}}{3 x + 11} + \ln (3 x + 11) \frac{d}{d x} [e^{x}]$

Step 4

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

$\frac{3 e^{x}}{3 x + 11} + \ln (3 x + 11) e^{x}$

Step 5

To write $\ln (3 x + 11) e^{x}$ as a fraction with a common denominator, multiply by $\frac{3 x + 11}{3 x + 11}$ .

$\frac{3 e^{x}}{3 x + 11} + \frac{\ln (3 x + 11) e^{x} (3 x + 11)}{3 x + 11}$

Step 6

Combine the numerators over the common denominator.

$\frac{3 e^{x} + \ln (3 x + 11) e^{x} (3 x + 11)}{3 x + 11}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

Apply the distributive property.

$\frac{3 e^{x} + \ln (3 x + 11) e^{x} (3 x) + \ln (3 x + 11) e^{x} \cdot 11}{3 x + 11}$

Step 7.1.1.2

Rewrite using the commutative property of multiplication.

$\frac{3 e^{x} + 3 \ln (3 x + 11) e^{x} x + \ln (3 x + 11) e^{x} \cdot 11}{3 x + 11}$

Step 7.1.1.3

Simplify $11 \ln (3 x + 11)$ by moving $11$ inside the logarithm.

$\frac{3 e^{x} + 3 \ln (3 x + 11) e^{x} x + \ln ({(3 x + 11)}^{11}) e^{x}}{3 x + 11}$

Step 7.1.1.4

Simplify $3 \ln (3 x + 11)$ by moving $3$ inside the logarithm.

$\frac{3 e^{x} + \ln ({(3 x + 11)}^{3}) e^{x} x + \ln ({(3 x + 11)}^{11}) e^{x}}{3 x + 11}$

Step 7.1.2

Reorder factors in $3 e^{x} + \ln ({(3 x + 11)}^{3}) e^{x} x + \ln ({(3 x + 11)}^{11}) e^{x}$ .

$\frac{3 e^{x} + x e^{x} \ln ({(3 x + 11)}^{3}) + e^{x} \ln ({(3 x + 11)}^{11})}{3 x + 11}$

Step 7.2

Reorder terms.

$\frac{e^{x} x \ln ({(3 x + 11)}^{3}) + e^{x} \ln ({(3 x + 11)}^{11}) + 3 e^{x}}{3 x + 11}$