Calculus Examples

$f (x) = \ln (x) \cdot \log_{5} (\sin (x))$

Step 1

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 $f (x) = \ln (x)$ and $g (x) = \log_{5} (\sin (x))$ .

$\ln (x) \frac{d}{d x} [\log_{5} (\sin (x))] + \log_{5} (\sin (x)) \frac{d}{d x} [\ln (x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{5} (x)$ and $g (x) = \sin (x)$ .

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Step 2.1

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

$\ln (x) (\frac{d}{d u} [\log_{5} (u)] \frac{d}{d x} [\sin (x)]) + \log_{5} (\sin (x)) \frac{d}{d x} [\ln (x)]$

Step 2.2

The derivative of $\log_{5} (u)$ with respect to $u$ is $\frac{1}{u \ln (5)}$ .

$\ln (x) (\frac{1}{u \ln (5)} \frac{d}{d x} [\sin (x)]) + \log_{5} (\sin (x)) \frac{d}{d x} [\ln (x)]$

Step 2.3

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

$\ln (x) (\frac{1}{\sin (x) \ln (5)} \frac{d}{d x} [\sin (x)]) + \log_{5} (\sin (x)) \frac{d}{d x} [\ln (x)]$

Step 3

Combine $\frac{1}{\sin (x) \ln (5)}$ and $\ln (x)$ .

$\frac{\ln (x)}{\sin (x) \ln (5)} \frac{d}{d x} [\sin (x)] + \log_{5} (\sin (x)) \frac{d}{d x} [\ln (x)]$

Step 4

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\frac{\ln (x)}{\sin (x) \ln (5)} \cos (x) + \log_{5} (\sin (x)) \frac{d}{d x} [\ln (x)]$

Step 5

Combine $\frac{\ln (x)}{\sin (x) \ln (5)}$ and $\cos (x)$ .

$\frac{\ln (x) \cos (x)}{\sin (x) \ln (5)} + \log_{5} (\sin (x)) \frac{d}{d x} [\ln (x)]$

Step 6

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

$\frac{\ln (x) \cos (x)}{\sin (x) \ln (5)} + \log_{5} (\sin (x)) \frac{1}{x}$

Step 7

Combine $\log_{5} (\sin (x))$ and $\frac{1}{x}$ .

$\frac{\ln (x) \cos (x)}{\sin (x) \ln (5)} + \frac{\log_{5} (\sin (x))}{x}$

Step 8

Simplify.

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Step 8.1

Reorder terms.

$\frac{\log_{5} (\sin (x))}{x} + \frac{\ln (x) \cos (x)}{\sin (x) \ln (5)}$

Step 8.2

Simplify each term.

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Step 8.2.1

Separate fractions.

$\frac{\log_{5} (\sin (x))}{x} + \frac{\ln (x)}{\ln (5)} \cdot \frac{\cos (x)}{\sin (x)}$

Step 8.2.2

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

$\frac{\log_{5} (\sin (x))}{x} + \frac{\ln (x)}{\ln (5)} \cot (x)$

Step 8.2.3

Combine $\frac{\ln (x)}{\ln (5)}$ and $\cot (x)$ .

$\frac{\log_{5} (\sin (x))}{x} + \frac{\ln (x) \cot (x)}{\ln (5)}$