Calculus Examples

$e^{6 \ln (x)}$

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 $g (x) = 6 \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $6 \ln (x)$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [6 \ln (x)]$

Step 1.2

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

$e^{u} \frac{d}{d x} [6 \ln (x)]$

Step 1.3

Replace all occurrences of $u$ with $6 \ln (x)$ .

$e^{6 \ln (x)} \frac{d}{d x} [6 \ln (x)]$

Step 2

Since $6$ is constant with respect to $x$ , the derivative of $6 \ln (x)$ with respect to $x$ is $6 \frac{d}{d x} [\ln (x)]$ .

$e^{6 \ln (x)} (6 \frac{d}{d x} [\ln (x)])$

Step 3

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

$e^{6 \ln (x)} (6 \frac{1}{x})$

Step 4

Combine fractions.

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Step 4.1

Combine $6$ and $\frac{1}{x}$ .

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

Step 4.2

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

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

Step 4.3

Move $6$ to the left of $e^{6 \ln (x)}$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Simplify $6 \ln (x)$ by moving $6$ inside the logarithm.

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

Step 5.1.2

Exponentiation and log are inverse functions.

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

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 $1$ .

$6 x^{5}$