Calculus Examples

$\ln (\sec (\frac{2}{x}))$

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (\frac{2}{x})}$ .

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

Step 4

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

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

Step 5

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

Tap for more steps...

Step 5.1

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

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

Step 5.2

The derivative of $\sec (u_{2})$ with respect to $u_{2}$ is $\sec (u_{2}) \tan (u_{2})$ .

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

Step 5.3

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

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

Step 6

Differentiate.

Tap for more steps...

Step 6.1

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 6.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 6.3

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

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

Step 6.4

Multiply $- 1$ by $2$ .

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

Step 7

Simplify.

Tap for more steps...

Step 7.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 7.2

Combine terms.

Tap for more steps...

Step 7.2.1

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

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

Step 7.2.2

Move the negative in front of the fraction.

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

Step 7.2.3

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

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

Step 7.2.4

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

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

Step 7.2.5

Combine $\tan (\frac{2}{x})$ and $\frac{\sec (\frac{2}{x}) (\cos (\frac{2}{x}) \cdot 2)}{x^{2}}$ .

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

Step 7.2.6

Move $2$ to the left of $\cos (\frac{2}{x})$ .

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

Step 7.2.7

Move $2$ to the left of $\sec (\frac{2}{x})$ .

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

Step 7.2.8

Move $2$ to the left of $\tan (\frac{2}{x})$ .

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

Step 7.3

Simplify the numerator.

Tap for more steps...

Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

Combine exponents.

Tap for more steps...

Step 7.3.3.1

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

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

Step 7.3.3.2

Multiply $\frac{2 \sin (\frac{2}{x})}{\cos (\frac{2}{x})}$ by $\frac{1}{\cos (\frac{2}{x})}$ .

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

Step 7.3.3.3

Raise $\cos (\frac{2}{x})$ to the power of $1$ .

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

Step 7.3.3.4

Raise $\cos (\frac{2}{x})$ to the power of $1$ .

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

Step 7.3.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.3.3.6

Add $1$ and $1$ .

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

Step 7.3.3.7

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

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

Step 7.3.4

Reduce the expression $\frac{2 \sin (\frac{2}{x}) \cos (\frac{2}{x})}{\cos^{2} (\frac{2}{x})}$ by cancelling the common factors.

Tap for more steps...

Step 7.3.4.1

Factor $\cos (\frac{2}{x})$ out of $2 \sin (\frac{2}{x}) \cos (\frac{2}{x})$ .

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

Step 7.3.4.2

Factor $\cos (\frac{2}{x})$ out of $\cos^{2} (\frac{2}{x})$ .

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

Step 7.3.4.3

Cancel the common factor.

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

Step 7.3.4.4

Rewrite the expression.

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

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.5

Combine.

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

Step 7.6

Multiply $2$ by $1$ .

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

Step 7.7

Separate fractions.

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

Step 7.8

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

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

Step 7.9

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

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