Calculus Examples

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

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 $g (x) = 3 \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $3 \ln (x)$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $3 \ln (x)$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

Exponentiation and log are inverse functions.

$\frac{3 x^{3}}{x}$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

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

$\frac{x (3 x^{2})}{x^{1}}$

Step 5.2.2.2

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

$\frac{x (3 x^{2})}{x \cdot 1}$

Step 5.2.2.3

Cancel the common factor.

$\frac{x (3 x^{2})}{x \cdot 1}$

Step 5.2.2.4

Rewrite the expression.

$\frac{3 x^{2}}{1}$

Step 5.2.2.5

Divide $3 x^{2}$ by $1$ .

$3 x^{2}$