Calculus Examples

$\ln (\cos (x^{2}))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \cos (x^{2})$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\cos (x^{2})$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\cos (x^{2})]$

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d x} [\cos (x^{2})]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\cos (x^{2})$ .

$\frac{1}{\cos (x^{2})} \frac{d}{d x} [\cos (x^{2})]$

Step 2

Convert from $\frac{1}{\cos (x^{2})}$ to $\sec (x^{2})$ .

$\sec (x^{2}) \frac{d}{d x} [\cos (x^{2})]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = x^{2}$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $x^{2}$ .

$\sec (x^{2}) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [x^{2}])$

Step 3.2

The derivative of $\cos (u_{2})$ with respect to $u_{2}$ is $- \sin (u_{2})$ .

$\sec (x^{2}) (- \sin (u_{2}) \frac{d}{d x} [x^{2}])$

Step 3.3

Replace all occurrences of $u_{2}$ with $x^{2}$ .

$\sec (x^{2}) (- \sin (x^{2}) \frac{d}{d x} [x^{2}])$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\sec (x^{2}) (- \sin (x^{2}) (2 x))$

Step 4.2

Multiply $2$ by $- 1$ .

$\sec (x^{2}) (- 2 \sin (x^{2}) x)$

Step 5

Simplify.

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Step 5.1

Reorder the factors of $\sec (x^{2}) (- 2 \sin (x^{2}) x)$ .

$- 2 x \sec (x^{2}) \sin (x^{2})$

Step 5.2

Rewrite $\sec (x^{2})$ in terms of sines and cosines.

$- 2 x \frac{1}{\cos (x^{2})} \sin (x^{2})$

Step 5.3

Multiply $- 2 x \frac{1}{\cos (x^{2})}$ .

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Step 5.3.1

Combine $- 2$ and $\frac{1}{\cos (x^{2})}$ .

$x \frac{- 2}{\cos (x^{2})} \sin (x^{2})$

Step 5.3.2

Combine $x$ and $\frac{- 2}{\cos (x^{2})}$ .

$\frac{x \cdot - 2}{\cos (x^{2})} \sin (x^{2})$

Step 5.4

Move $- 2$ to the left of $x$ .

$\frac{- 2 \cdot x}{\cos (x^{2})} \sin (x^{2})$

Step 5.5

Move the negative in front of the fraction.

$- \frac{2 \cdot x}{\cos (x^{2})} \sin (x^{2})$

Step 5.6

Combine $\sin (x^{2})$ and $\frac{2 x}{\cos (x^{2})}$ .

$- \frac{\sin (x^{2}) (2 x)}{\cos (x^{2})}$

Step 5.7

Move $2$ to the left of $\sin (x^{2})$ .

$- \frac{2 \cdot \sin (x^{2}) x}{\cos (x^{2})}$

Step 5.8

Separate fractions.

$- (\frac{2 x}{1} \cdot \frac{\sin (x^{2})}{\cos (x^{2})})$

Step 5.9

Convert from $\frac{\sin (x^{2})}{\cos (x^{2})}$ to $\tan (x^{2})$ .

$- (\frac{2 x}{1} \tan (x^{2}))$

Step 5.10

Divide $2 x$ by $1$ .

$- (2 x \tan (x^{2}))$

Step 5.11

Multiply $2$ by $- 1$ .

$- 2 x \tan (x^{2})$