Calculus Examples

$e^{4 x \ln (x^{3})}$

Step 1

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 $g (x) = 4 x \ln (x^{3})$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $4 x \ln (x^{3})$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [4 x \ln (x^{3})]$

Step 1.2

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

$e^{u_{1}} \frac{d}{d x} [4 x \ln (x^{3})]$

Step 1.3

Replace all occurrences of $u_{1}$ with $4 x \ln (x^{3})$ .

$e^{4 x \ln (x^{3})} \frac{d}{d x} [4 x \ln (x^{3})]$

Step 2

Since $4$ is constant with respect to $x$ , the derivative of $4 x \ln (x^{3})$ with respect to $x$ is $4 \frac{d}{d x} [x \ln (x^{3})]$ .

$e^{4 x \ln (x^{3})} (4 \frac{d}{d x} [x \ln (x^{3})])$

Step 3

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

$e^{4 x \ln (x^{3})} (4 (x \frac{d}{d x} [\ln (x^{3})] + \ln (x^{3}) \frac{d}{d x} [x]))$

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 $g (x) = x^{3}$ .

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Step 4.1

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

$e^{4 x \ln (x^{3})} (4 (x (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [x^{3}]) + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 4.2

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

$e^{4 x \ln (x^{3})} (4 (x (\frac{1}{u_{2}} \frac{d}{d x} [x^{3}]) + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 4.3

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

$e^{4 x \ln (x^{3})} (4 (x (\frac{1}{x^{3}} \frac{d}{d x} [x^{3}]) + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

$e^{4 x \ln (x^{3})} (4 (\frac{x}{x^{3}} \frac{d}{d x} [x^{3}] + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.2

Cancel the common factor of $x$ and $x^{3}$ .

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Step 5.2.1

Raise $x$ to the power of $1$ .

$e^{4 x \ln (x^{3})} (4 (\frac{x^{1}}{x^{3}} \frac{d}{d x} [x^{3}] + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.2.2

Factor $x$ out of $x^{1}$ .

$e^{4 x \ln (x^{3})} (4 (\frac{x \cdot 1}{x^{3}} \frac{d}{d x} [x^{3}] + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.2.3

Cancel the common factors.

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Step 5.2.3.1

Factor $x$ out of $x^{3}$ .

$e^{4 x \ln (x^{3})} (4 (\frac{x \cdot 1}{x \cdot x^{2}} \frac{d}{d x} [x^{3}] + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.2.3.2

Cancel the common factor.

$e^{4 x \ln (x^{3})} (4 (\frac{x \cdot 1}{x \cdot x^{2}} \frac{d}{d x} [x^{3}] + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.2.3.3

Rewrite the expression.

$e^{4 x \ln (x^{3})} (4 (\frac{1}{x^{2}} \frac{d}{d x} [x^{3}] + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$e^{4 x \ln (x^{3})} (4 (\frac{1}{x^{2}} (3 x^{2}) + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.4

Simplify terms.

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Step 5.4.1

Combine $3$ and $\frac{1}{x^{2}}$ .

$e^{4 x \ln (x^{3})} (4 (\frac{3}{x^{2}} x^{2} + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.4.2

Combine $\frac{3}{x^{2}}$ and $x^{2}$ .

$e^{4 x \ln (x^{3})} (4 (\frac{3 x^{2}}{x^{2}} + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.4.3

Cancel the common factor of $x^{2}$ .

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Step 5.4.3.1

Cancel the common factor.

$e^{4 x \ln (x^{3})} (4 (\frac{3 x^{2}}{x^{2}} + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.4.3.2

Divide $3$ by $1$ .

$e^{4 x \ln (x^{3})} (4 (3 + \ln (x^{3}) \frac{d}{d x} [x]))$

Step 5.5

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

$e^{4 x \ln (x^{3})} (4 (3 + \ln (x^{3}) \cdot 1))$

Step 5.6

Multiply $\ln (x^{3})$ by $1$ .

$e^{4 x \ln (x^{3})} (4 (3 + \ln (x^{3})))$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$e^{4 x \ln (x^{3})} (4 \cdot 3 + 4 \ln (x^{3}))$

Step 6.2

Apply the distributive property.

$e^{4 x \ln (x^{3})} (4 \cdot 3) + e^{4 x \ln (x^{3})} (4 \ln (x^{3}))$

Step 6.3

Combine terms.

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Step 6.3.1

Multiply $4$ by $3$ .

$e^{4 x \ln (x^{3})} \cdot 12 + e^{4 x \ln (x^{3})} (4 \ln (x^{3}))$

Step 6.3.2

Move $12$ to the left of $e^{4 x \ln (x^{3})}$ .

$12 e^{4 x \ln (x^{3})} + e^{4 x \ln (x^{3})} (4 \ln (x^{3}))$

Step 6.4

Reorder terms.

$4 e^{4 x \ln (x^{3})} \ln (x^{3}) + 12 e^{4 x \ln (x^{3})}$