Calculus Examples

$f (x) = \sin (9 \ln (x))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 9 \ln (x)$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $9 \ln (x)$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [9 \ln (x)]$

Step 1.2

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

$\cos (u) \frac{d}{d x} [9 \ln (x)]$

Step 1.3

Replace all occurrences of $u$ with $9 \ln (x)$ .

$\cos (9 \ln (x)) \frac{d}{d x} [9 \ln (x)]$

Step 2

Since $9$ is constant with respect to $x$ , the derivative of $9 \ln (x)$ with respect to $x$ is $9 \frac{d}{d x} [\ln (x)]$ .

$\cos (9 \ln (x)) (9 \frac{d}{d x} [\ln (x)])$

Step 3

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

$\cos (9 \ln (x)) (9 \frac{1}{x})$

Step 4

Combine fractions.

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Step 4.1

Combine $9$ and $\frac{1}{x}$ .

$\cos (9 \ln (x)) \frac{9}{x}$

Step 4.2

Combine $\cos (9 \ln (x))$ and $\frac{9}{x}$ .

$\frac{\cos (9 \ln (x)) \cdot 9}{x}$

Step 4.3

Move $9$ to the left of $\cos (9 \ln (x))$ .

$\frac{9 \cdot \cos (9 \ln (x))}{x}$

Step 5

Simplify $9 \ln (x)$ by moving $9$ inside the logarithm.

$\frac{9 \cos (\ln (x^{9}))}{x}$