Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite ${\tan (x)}^{\frac{1}{x}}$ as $e^{\ln ({\tan (x)}^{\frac{1}{x}})}$ .

$\frac{d}{d x} [e^{\ln ({\tan (x)}^{\frac{1}{x}})}]$

Step 1.2

Expand $\ln ({\tan (x)}^{\frac{1}{x}})$ by moving $\frac{1}{x}$ outside the logarithm.

$\frac{d}{d x} [e^{\frac{1}{x} \ln (\tan (x))}]$

Step 2

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

$\frac{d}{d x} [e^{\frac{\ln (\tan (x))}{x}}]$

Step 3

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

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Step 3.1

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

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [\frac{\ln (\tan (x))}{x}]$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [\frac{\ln (\tan (x))}{x}]$

Step 3.3

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

$e^{\frac{\ln (\tan (x))}{x}} \frac{d}{d x} [\frac{\ln (\tan (x))}{x}]$

Step 4

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \ln (\tan (x))$ and $g (x) = x$ .

$e^{\frac{\ln (\tan (x))}{x}} \frac{x \frac{d}{d x} [\ln (\tan (x))] - \ln (\tan (x)) \frac{d}{d x} [x]}{x^{2}}$

Step 5

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

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Step 5.1

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

$e^{\frac{\ln (\tan (x))}{x}} \frac{x (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [\tan (x)]) - \ln (\tan (x)) \frac{d}{d x} [x]}{x^{2}}$

Step 5.2

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

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

Step 5.3

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

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

Step 6

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 7

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

$e^{\frac{\ln (\tan (x))}{x}} \frac{x (\frac{\cos (x)}{\sin (x)} \frac{d}{d x} [\tan (x)]) - \ln (\tan (x)) \frac{d}{d x} [x]}{x^{2}}$

Step 8

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

$e^{\frac{\ln (\tan (x))}{x}} \frac{x (\cot (x) \frac{d}{d x} [\tan (x)]) - \ln (\tan (x)) \frac{d}{d x} [x]}{x^{2}}$

Step 9

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

$e^{\frac{\ln (\tan (x))}{x}} \frac{x (\cot (x) \sec^{2} (x)) - \ln (\tan (x)) \frac{d}{d x} [x]}{x^{2}}$

Step 10

Differentiate using the Power Rule.

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Step 10.1

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

$e^{\frac{\ln (\tan (x))}{x}} \frac{x (\cot (x) \sec^{2} (x)) - \ln (\tan (x)) \cdot 1}{x^{2}}$

Step 10.2

Combine fractions.

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Step 10.2.1

Multiply $- 1$ by $1$ .

$e^{\frac{\ln (\tan (x))}{x}} \frac{x (\cot (x) \sec^{2} (x)) - \ln (\tan (x))}{x^{2}}$

Step 10.2.2

Combine $e^{\frac{\ln (\tan (x))}{x}}$ and $\frac{x (\cot (x) \sec^{2} (x)) - \ln (\tan (x))}{x^{2}}$ .

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\cot (x) \sec^{2} (x)) - \ln (\tan (x)))}{x^{2}}$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\cot (x) \sec^{2} (x))) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\frac{\cos (x)}{\sin (x)} \sec^{2} (x))) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 11.2.1.3

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\frac{\cos (x)}{\sin (x)} \cdot \frac{1^{2}}{\cos^{2} (x)})) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.4

Cancel the common factor of $\cos (x)$ .

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Step 11.2.1.4.1

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

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\frac{\cos (x)}{\sin (x)} \cdot \frac{1^{2}}{\cos (x) \cos (x)})) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.4.2

Cancel the common factor.

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\frac{\cos (x)}{\sin (x)} \cdot \frac{1^{2}}{\cos (x) \cos (x)})) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.4.3

Rewrite the expression.

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\frac{1}{\sin (x)} \cdot \frac{1^{2}}{\cos (x)})) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.5

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

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

Step 11.2.1.6

Separate fractions.

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\frac{1^{2}}{\cos (x)} \cdot \frac{1}{\sin (x)})) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.7

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\frac{1^{2}}{\cos (x)} \csc (x))) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.8

Separate fractions.

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\frac{1^{2}}{1} \cdot \frac{1}{\cos (x)} \csc (x))) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.9

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

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x (\frac{1^{2}}{1} \sec (x) \csc (x))) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.10

Rewrite using the commutative property of multiplication.

$\frac{\frac{1^{2}}{1} e^{\frac{\ln (\tan (x))}{x}} x \sec (x) \csc (x) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.11

Divide $1^{2}$ by $1$ .

$\frac{1^{2} e^{\frac{\ln (\tan (x))}{x}} x \sec (x) \csc (x) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.12

One to any power is one.

$\frac{1 e^{\frac{\ln (\tan (x))}{x}} x \sec (x) \csc (x) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.13

Multiply $e^{\frac{\ln (\tan (x))}{x}}$ by $1$ .

$\frac{e^{\frac{\ln (\tan (x))}{x}} x \sec (x) \csc (x) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.2.1.14

Rewrite using the commutative property of multiplication.

$\frac{e^{\frac{\ln (\tan (x))}{x}} x \sec (x) \csc (x) - e^{\frac{\ln (\tan (x))}{x}} \ln (\tan (x))}{x^{2}}$

Step 11.2.2

Reorder factors in $e^{\frac{\ln (\tan (x))}{x}} x \sec (x) \csc (x) - e^{\frac{\ln (\tan (x))}{x}} \ln (\tan (x))$ .

$\frac{x e^{\frac{\ln (\tan (x))}{x}} \sec (x) \csc (x) - e^{\frac{\ln (\tan (x))}{x}} \ln (\tan (x))}{x^{2}}$

Step 11.3

Reorder terms.

$\frac{e^{\frac{\ln (\tan (x))}{x}} x \csc (x) \sec (x) - e^{\frac{\ln (\tan (x))}{x}} \ln (\tan (x))}{x^{2}}$

Step 11.4

Factor $e^{\frac{\ln (\tan (x))}{x}}$ out of $e^{\frac{\ln (\tan (x))}{x}} x \csc (x) \sec (x) - e^{\frac{\ln (\tan (x))}{x}} \ln (\tan (x))$ .

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Step 11.4.1

Reorder $e^{\frac{\ln (\tan (x))}{x}}$ and $x$ .

$\frac{x e^{\frac{\ln (\tan (x))}{x}} \csc (x) \sec (x) - e^{\frac{\ln (\tan (x))}{x}} \ln (\tan (x))}{x^{2}}$

Step 11.4.2

Factor $e^{\frac{\ln (\tan (x))}{x}}$ out of $x e^{\frac{\ln (\tan (x))}{x}} \csc (x) \sec (x)$ .

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x \csc (x) \sec (x)) - e^{\frac{\ln (\tan (x))}{x}} \ln (\tan (x))}{x^{2}}$

Step 11.4.3

Factor $e^{\frac{\ln (\tan (x))}{x}}$ out of $- e^{\frac{\ln (\tan (x))}{x}} \ln (\tan (x))$ .

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x \csc (x) \sec (x)) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))}{x^{2}}$

Step 11.4.4

Factor $e^{\frac{\ln (\tan (x))}{x}}$ out of $e^{\frac{\ln (\tan (x))}{x}} (x \csc (x) \sec (x)) + e^{\frac{\ln (\tan (x))}{x}} (- \ln (\tan (x)))$ .

$\frac{e^{\frac{\ln (\tan (x))}{x}} (x \csc (x) \sec (x) - \ln (\tan (x)))}{x^{2}}$