Calculus Examples

$e^{3 \ln (x^{2})}$

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 $g (x) = 3 \ln (x^{2})$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $3 \ln (x^{2})$ .

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

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 $a$ = $e$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with $3 \ln (x^{2})$ .

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

Step 2

Since $3$ is constant with respect to $x$ , the derivative of $3 \ln (x^{2})$ with respect to $x$ is $3 \frac{d}{d x} [\ln (x^{2})]$ .

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

Step 3

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 $g (x) = x^{2}$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

Combine $\frac{3}{x^{2}}$ and $e^{3 \ln (x^{2})}$ .

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

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{3 e^{3 \ln (x^{2})}}{x^{2}} (2 x)$

Step 4.4

Simplify terms.

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Step 4.4.1

Combine $2$ and $\frac{3 e^{3 \ln (x^{2})}}{x^{2}}$ .

$\frac{2 (3 e^{3 \ln (x^{2})})}{x^{2}} x$

Step 4.4.2

Multiply $3$ by $2$ .

$\frac{6 e^{3 \ln (x^{2})}}{x^{2}} x$

Step 4.4.3

Combine $\frac{6 e^{3 \ln (x^{2})}}{x^{2}}$ and $x$ .

$\frac{6 e^{3 \ln (x^{2})} x}{x^{2}}$

Step 4.4.4

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

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Step 4.4.4.1

Factor $x$ out of $6 e^{3 \ln (x^{2})} x$ .

$\frac{x (6 e^{3 \ln (x^{2})})}{x^{2}}$

Step 4.4.4.2

Cancel the common factors.

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Step 4.4.4.2.1

Factor $x$ out of $x^{2}$ .

$\frac{x (6 e^{3 \ln (x^{2})})}{x \cdot x}$

Step 4.4.4.2.2

Cancel the common factor.

$\frac{x (6 e^{3 \ln (x^{2})})}{x \cdot x}$

Step 4.4.4.2.3

Rewrite the expression.

$\frac{6 e^{3 \ln (x^{2})}}{x}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Simplify $3 \ln (x^{2})$ by moving $3$ inside the logarithm.

$\frac{6 e^{\ln ({(x^{2})}^{3})}}{x}$

Step 5.1.2

Exponentiation and log are inverse functions.

$\frac{6 {(x^{2})}^{3}}{x}$

Step 5.1.3

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 5.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{6 x^{2 \cdot 3}}{x}$

Step 5.1.3.2

Multiply $2$ by $3$ .

$\frac{6 x^{6}}{x}$

Step 5.2

Cancel the common factor of $x^{6}$ and $x$ .

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Step 5.2.1

Factor $x$ out of $6 x^{6}$ .

$\frac{x (6 x^{5})}{x}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

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

$\frac{x (6 x^{5})}{x^{1}}$

Step 5.2.2.2

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

$\frac{x (6 x^{5})}{x \cdot 1}$

Step 5.2.2.3

Cancel the common factor.

$\frac{x (6 x^{5})}{x \cdot 1}$

Step 5.2.2.4

Rewrite the expression.

$\frac{6 x^{5}}{1}$

Step 5.2.2.5

Divide $6 x^{5}$ by $1$ .

$6 x^{5}$