Calculus Examples

$f (x) = e^{\log (\sin (3 x))}$

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 $g (x) = \log (\sin (3 x))$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\log (\sin (3 x))$ .

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

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 $a$ = $e$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\log (\sin (3 x))$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log (x)$ and $g (x) = \sin (3 x)$ .

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Step 2.1

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

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

Step 2.2

The derivative of $\log (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2} \ln (10)}$ .

$e^{\log (\sin (3 x))} (\frac{1}{u_{2} \ln (10)} \frac{d}{d x} [\sin (3 x)])$

Step 2.3

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

$e^{\log (\sin (3 x))} (\frac{1}{\sin (3 x) \ln (10)} \frac{d}{d x} [\sin (3 x)])$

Step 3

Combine $\frac{1}{\sin (3 x) \ln (10)}$ and $e^{\log (\sin (3 x))}$ .

$\frac{e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)} \frac{d}{d x} [\sin (3 x)]$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 3 x$ .

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Step 4.1

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

$\frac{e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)} (\frac{d}{d u_{3}} [\sin (u_{3})] \frac{d}{d x} [3 x])$

Step 4.2

The derivative of $\sin (u_{3})$ with respect to $u_{3}$ is $\cos (u_{3})$ .

$\frac{e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)} (\cos (u_{3}) \frac{d}{d x} [3 x])$

Step 4.3

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

$\frac{e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)} (\cos (3 x) \frac{d}{d x} [3 x])$

Step 5

Differentiate.

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Step 5.1

Combine $\cos (3 x)$ and $\frac{e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)}$ .

$\frac{\cos (3 x) e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)} \frac{d}{d x} [3 x]$

Step 5.2

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

$\frac{\cos (3 x) e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)} (3 \frac{d}{d x} [x])$

Step 5.3

Combine $3$ and $\frac{\cos (3 x) e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)}$ .

$\frac{3 (\cos (3 x) e^{\log (\sin (3 x))})}{\sin (3 x) \ln (10)} \frac{d}{d x} [x]$

Step 5.4

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

$\frac{3 \cos (3 x) e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)} \cdot 1$

Step 5.5

Multiply $\frac{3 \cos (3 x) e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)}$ by $1$ .

$\frac{3 \cos (3 x) e^{\log (\sin (3 x))}}{\sin (3 x) \ln (10)}$

Step 6

Simplify.

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Step 6.1

Reorder factors in $3 \cos (3 x) e^{\log (\sin (3 x))}$ .

$\frac{3 e^{\log (\sin (3 x))} \cos (3 x)}{\sin (3 x) \ln (10)}$

Step 6.2

Separate fractions.

$\frac{3 e^{\log (\sin (3 x))}}{\ln (10)} \cdot \frac{\cos (3 x)}{\sin (3 x)}$

Step 6.3

Convert from $\frac{\cos (3 x)}{\sin (3 x)}$ to $\cot (3 x)$ .

$\frac{3 e^{\log (\sin (3 x))}}{\ln (10)} \cot (3 x)$

Step 6.4

Combine $\frac{3 e^{\log (\sin (3 x))}}{\ln (10)}$ and $\cot (3 x)$ .

$\frac{3 e^{\log (\sin (3 x))} \cot (3 x)}{\ln (10)}$