Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite ${\cos (x)}^{\frac{1}{x^{2}}}$ as $e^{\ln ({\cos (x)}^{\frac{1}{x^{2}}})}$ .

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

Step 1.2

Expand $\ln ({\cos (x)}^{\frac{1}{x^{2}}})$ by moving $\frac{1}{x^{2}}$ outside the logarithm.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

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 $a$ = $e$ .

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

Step 3.3

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

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

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 $f (x) = \ln (\cos (x))$ and $g (x) = x^{2}$ .

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

Step 5

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.2

Multiply $2$ by $2$ .

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x^{2} \frac{d}{d x} [\ln (\cos (x))] - \ln (\cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 6

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 $g (x) = \cos (x)$ .

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Step 6.1

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

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x^{2} (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [\cos (x)]) - \ln (\cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 6.2

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

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x^{2} (\frac{1}{u_{2}} \frac{d}{d x} [\cos (x)]) - \ln (\cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 6.3

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

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x^{2} (\frac{1}{\cos (x)} \frac{d}{d x} [\cos (x)]) - \ln (\cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 7

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

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x^{2} (\sec (x) \frac{d}{d x} [\cos (x)]) - \ln (\cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 8

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

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x^{2} (\sec (x) (- \sin (x))) - \ln (\cos (x)) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 9

Differentiate using the Power Rule.

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Step 9.1

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

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x^{2} (\sec (x) (- \sin (x))) - \ln (\cos (x)) (2 x)}{x^{4}}$

Step 9.2

Simplify with factoring out.

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Step 9.2.1

Multiply $2$ by $- 1$ .

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x^{2} (\sec (x) (- \sin (x))) - 2 \ln (\cos (x)) x}{x^{4}}$

Step 9.2.2

Factor $x$ out of $x^{2} (\sec (x) (- \sin (x))) - 2 \ln (\cos (x)) x$ .

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Step 9.2.2.1

Factor $x$ out of $x^{2} (\sec (x) (- \sin (x)))$ .

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x (x (\sec (x) (- \sin (x)))) - 2 \ln (\cos (x)) x}{x^{4}}$

Step 9.2.2.2

Factor $x$ out of $- 2 \ln (\cos (x)) x$ .

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x (x (\sec (x) (- \sin (x)))) + x (- 2 \ln (\cos (x)))}{x^{4}}$

Step 9.2.2.3

Factor $x$ out of $x (x (\sec (x) (- \sin (x)))) + x (- 2 \ln (\cos (x)))$ .

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x (x (\sec (x) (- \sin (x))) - 2 \ln (\cos (x)))}{x^{4}}$

Step 10

Cancel the common factors.

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Step 10.1

Factor $x$ out of $x^{4}$ .

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x (x (\sec (x) (- \sin (x))) - 2 \ln (\cos (x)))}{x \cdot x^{3}}$

Step 10.2

Cancel the common factor.

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x (x (\sec (x) (- \sin (x))) - 2 \ln (\cos (x)))}{x \cdot x^{3}}$

Step 10.3

Rewrite the expression.

$e^{\frac{\ln (\cos (x))}{x^{2}}} \frac{x (\sec (x) (- \sin (x))) - 2 \ln (\cos (x))}{x^{3}}$

Step 11

Combine $e^{\frac{\ln (\cos (x))}{x^{2}}}$ and $\frac{x (\sec (x) (- \sin (x))) - 2 \ln (\cos (x))}{x^{3}}$ .

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (x (\sec (x) (- \sin (x))) - 2 \ln (\cos (x)))}{x^{3}}$

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (x (\sec (x) (- \sin (x)))) + e^{\frac{\ln (\cos (x))}{x^{2}}} (- 2 \ln (\cos (x)))}{x^{3}}$

Step 12.2

Simplify the numerator.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{- e^{\frac{\ln (\cos (x))}{x^{2}}} x \sec (x) \sin (x) + e^{\frac{\ln (\cos (x))}{x^{2}}} (- 2 \ln (\cos (x)))}{x^{3}}$

Step 12.2.1.2

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

$\frac{- e^{\frac{\ln (\cos (x))}{x^{2}}} x \frac{1}{\cos (x)} \sin (x) + e^{\frac{\ln (\cos (x))}{x^{2}}} (- 2 \ln (\cos (x)))}{x^{3}}$

Step 12.2.1.3

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

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Step 12.2.1.3.1

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

$\frac{- x \frac{e^{\frac{\ln (\cos (x))}{x^{2}}}}{\cos (x)} \sin (x) + e^{\frac{\ln (\cos (x))}{x^{2}}} (- 2 \ln (\cos (x)))}{x^{3}}$

Step 12.2.1.3.2

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

$\frac{- \frac{e^{\frac{\ln (\cos (x))}{x^{2}}} x}{\cos (x)} \sin (x) + e^{\frac{\ln (\cos (x))}{x^{2}}} (- 2 \ln (\cos (x)))}{x^{3}}$

Step 12.2.1.4

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

$\frac{- \frac{\sin (x) (e^{\frac{\ln (\cos (x))}{x^{2}}} x)}{\cos (x)} + e^{\frac{\ln (\cos (x))}{x^{2}}} (- 2 \ln (\cos (x)))}{x^{3}}$

Step 12.2.1.5

Rewrite using the commutative property of multiplication.

$\frac{- \frac{\sin (x) e^{\frac{\ln (\cos (x))}{x^{2}}} x}{\cos (x)} - 2 e^{\frac{\ln (\cos (x))}{x^{2}}} \ln (\cos (x))}{x^{3}}$

Step 12.2.1.6

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

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Step 12.2.1.6.1

Reorder $\ln (\cos (x))$ and $- 2$ .

$\frac{- \frac{\sin (x) e^{\frac{\ln (\cos (x))}{x^{2}}} x}{\cos (x)} - 2 \cdot \ln (\cos (x)) e^{\frac{\ln (\cos (x))}{x^{2}}}}{x^{3}}$

Step 12.2.1.6.2

Simplify $- 2 \cdot \ln (\cos (x))$ by moving $2$ inside the logarithm.

$\frac{- \frac{\sin (x) e^{\frac{\ln (\cos (x))}{x^{2}}} x}{\cos (x)} - \ln (\cos^{2} (x)) e^{\frac{\ln (\cos (x))}{x^{2}}}}{x^{3}}$

Step 12.2.2

Simplify each term.

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Step 12.2.2.1

Separate fractions.

$\frac{- (\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} x}{1} \cdot \frac{\sin (x)}{\cos (x)}) - \ln (\cos^{2} (x)) e^{\frac{\ln (\cos (x))}{x^{2}}}}{x^{3}}$

Step 12.2.2.2

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

$\frac{- (\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} x}{1} \tan (x)) - \ln (\cos^{2} (x)) e^{\frac{\ln (\cos (x))}{x^{2}}}}{x^{3}}$

Step 12.2.2.3

Divide $e^{\frac{\ln (\cos (x))}{x^{2}}} x$ by $1$ .

$\frac{- e^{\frac{\ln (\cos (x))}{x^{2}}} x \tan (x) - \ln (\cos^{2} (x)) e^{\frac{\ln (\cos (x))}{x^{2}}}}{x^{3}}$

Step 12.2.3

Reorder factors in $- e^{\frac{\ln (\cos (x))}{x^{2}}} x \tan (x) - \ln (\cos^{2} (x)) e^{\frac{\ln (\cos (x))}{x^{2}}}$ .

$\frac{- x e^{\frac{\ln (\cos (x))}{x^{2}}} \tan (x) - e^{\frac{\ln (\cos (x))}{x^{2}}} \ln (\cos^{2} (x))}{x^{3}}$

Step 12.3

Reorder terms.

$\frac{- e^{\frac{\ln (\cos (x))}{x^{2}}} x \tan (x) - e^{\frac{\ln (\cos (x))}{x^{2}}} \ln (\cos^{2} (x))}{x^{3}}$

Step 12.4

Simplify the numerator.

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Step 12.4.1

Factor $e^{\frac{\ln (\cos (x))}{x^{2}}}$ out of $- e^{\frac{\ln (\cos (x))}{x^{2}}} x \tan (x) - e^{\frac{\ln (\cos (x))}{x^{2}}} \ln (\cos^{2} (x))$ .

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Step 12.4.1.1

Move $e^{\frac{\ln (\cos (x))}{x^{2}}}$ .

$\frac{- x e^{\frac{\ln (\cos (x))}{x^{2}}} \tan (x) - e^{\frac{\ln (\cos (x))}{x^{2}}} \ln (\cos^{2} (x))}{x^{3}}$

Step 12.4.1.2

Factor $e^{\frac{\ln (\cos (x))}{x^{2}}}$ out of $- x e^{\frac{\ln (\cos (x))}{x^{2}}} \tan (x)$ .

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (- x \tan (x)) - e^{\frac{\ln (\cos (x))}{x^{2}}} \ln (\cos^{2} (x))}{x^{3}}$

Step 12.4.1.3

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

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (- x \tan (x)) + e^{\frac{\ln (\cos (x))}{x^{2}}} (- \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.4.1.4

Factor $e^{\frac{\ln (\cos (x))}{x^{2}}}$ out of $e^{\frac{\ln (\cos (x))}{x^{2}}} (- x \tan (x)) + e^{\frac{\ln (\cos (x))}{x^{2}}} (- \ln (\cos^{2} (x)))$ .

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (- x \tan (x) - \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.4.2

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

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (- x \frac{\sin (x)}{\cos (x)} - \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.4.3

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

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (- \frac{\sin (x) x}{\cos (x)} - \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.4.4

Factor $- 1$ out of $- \frac{\sin (x) x}{\cos (x)} - \ln (\cos^{2} (x))$ .

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Step 12.4.4.1

Reorder $\sin (x)$ and $x$ .

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (- \frac{x \sin (x)}{\cos (x)} - \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.4.4.2

Factor $- 1$ out of $- \frac{x \sin (x)}{\cos (x)}$ .

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (- (\frac{x \sin (x)}{\cos (x)}) - \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.4.4.3

Factor $- 1$ out of $- \ln (\cos^{2} (x))$ .

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (- (\frac{x \sin (x)}{\cos (x)}) - (\ln (\cos^{2} (x))))}{x^{3}}$

Step 12.4.4.4

Factor $- 1$ out of $- (\frac{x \sin (x)}{\cos (x)}) - (\ln (\cos^{2} (x)))$ .

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (- (\frac{x \sin (x)}{\cos (x)} + \ln (\cos^{2} (x))))}{x^{3}}$

$\frac{e^{\frac{\ln (\cos (x))}{x^{2}}} \cdot - 1 (\frac{x \sin (x)}{\cos (x)} + \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.4.5

Factor out negative.

$\frac{- e^{\frac{\ln (\cos (x))}{x^{2}}} (\frac{x \sin (x)}{\cos (x)} + \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.5

Move the negative in front of the fraction.

$- \frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (\frac{x \sin (x)}{\cos (x)} + \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.6

Simplify each term.

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Step 12.6.1

Separate fractions.

$- \frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (\frac{x}{1} \cdot \frac{\sin (x)}{\cos (x)} + \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.6.2

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

$- \frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (\frac{x}{1} \tan (x) + \ln (\cos^{2} (x)))}{x^{3}}$

Step 12.6.3

Divide $x$ by $1$ .

$- \frac{e^{\frac{\ln (\cos (x))}{x^{2}}} (x \tan (x) + \ln (\cos^{2} (x)))}{x^{3}}$