Calculus Examples

$\ln (3 x) e^{5 x}$

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 $f (x) = \ln (3 x)$ and $g (x) = e^{5 x}$ .

$\ln (3 x) \frac{d}{d x} [e^{5 x}] + e^{5 x} \frac{d}{d x} [\ln (3 x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 5 x$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $5 x$ .

$\ln (3 x) (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [5 x]) + e^{5 x} \frac{d}{d x} [\ln (3 x)]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$\ln (3 x) (e^{u_{1}} \frac{d}{d x} [5 x]) + e^{5 x} \frac{d}{d x} [\ln (3 x)]$

Step 2.3

Replace all occurrences of $u_{1}$ with $5 x$ .

$\ln (3 x) (e^{5 x} \frac{d}{d x} [5 x]) + e^{5 x} \frac{d}{d x} [\ln (3 x)]$

Step 3

Differentiate.

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\ln (3 x) (e^{5 x} (5 \frac{d}{d x} [x])) + e^{5 x} \frac{d}{d x} [\ln (3 x)]$

Step 3.2

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

$\ln (3 x) (e^{5 x} (5 \cdot 1)) + e^{5 x} \frac{d}{d x} [\ln (3 x)]$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $5$ by $1$ .

$\ln (3 x) (e^{5 x} \cdot 5) + e^{5 x} \frac{d}{d x} [\ln (3 x)]$

Step 3.3.2

Move $5$ to the left of $e^{5 x}$ .

$\ln (3 x) (5 \cdot e^{5 x}) + e^{5 x} \frac{d}{d x} [\ln (3 x)]$

Step 4

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 $g (x) = 3 x$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $3 x$ .

$\ln (3 x) (5 e^{5 x}) + e^{5 x} (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [3 x])$

Step 4.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$\ln (3 x) (5 e^{5 x}) + e^{5 x} (\frac{1}{u_{2}} \frac{d}{d x} [3 x])$

Step 4.3

Replace all occurrences of $u_{2}$ with $3 x$ .

$\ln (3 x) (5 e^{5 x}) + e^{5 x} (\frac{1}{3 x} \frac{d}{d x} [3 x])$

Step 5

Differentiate.

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Step 5.1

Combine $\frac{1}{3 x}$ and $e^{5 x}$ .

$\ln (3 x) (5 e^{5 x}) + \frac{e^{5 x}}{3 x} \frac{d}{d x} [3 x]$

Step 5.2

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

$\ln (3 x) (5 e^{5 x}) + \frac{e^{5 x}}{3 x} (3 \frac{d}{d x} [x])$

Step 5.3

Simplify terms.

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Step 5.3.1

Combine $3$ and $\frac{e^{5 x}}{3 x}$ .

$\ln (3 x) (5 e^{5 x}) + \frac{3 e^{5 x}}{3 x} \frac{d}{d x} [x]$

Step 5.3.2

Cancel the common factor of $3$ .

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Step 5.3.2.1

Cancel the common factor.

$\ln (3 x) (5 e^{5 x}) + \frac{3 e^{5 x}}{3 x} \frac{d}{d x} [x]$

Step 5.3.2.2

Rewrite the expression.

$\ln (3 x) (5 e^{5 x}) + \frac{e^{5 x}}{x} \frac{d}{d x} [x]$

Step 5.4

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

$\ln (3 x) (5 e^{5 x}) + \frac{e^{5 x}}{x} \cdot 1$

Step 5.5

Simplify the expression.

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Step 5.5.1

Multiply $\frac{e^{5 x}}{x}$ by $1$ .

$\ln (3 x) (5 e^{5 x}) + \frac{e^{5 x}}{x}$

Step 5.5.2

Reorder terms.

$5 e^{5 x} \ln (3 x) + \frac{e^{5 x}}{x}$