Calculus Examples

$g (x) = \cos^{2} (\ln (\frac{1}{x})) + \frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}$

Step 1

By the Sum Rule, the derivative of $\cos^{2} (\ln (\frac{1}{x})) + \frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}$ with respect to $x$ is $\frac{d}{d x} [\cos^{2} (\ln (\frac{1}{x}))] + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$ .

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

Step 2

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

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (\ln (\frac{1}{x}))$ .

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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 $g (x) = \ln (\frac{1}{x})$ .

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Step 2.2.1

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

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

Step 2.2.2

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

$2 \cos (\ln (\frac{1}{x})) (- \sin (u_{2}) \frac{d}{d x} [\ln (\frac{1}{x})]) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.2.3

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

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) \frac{d}{d x} [\ln (\frac{1}{x})]) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{1}{x}$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{3}$ as $\frac{1}{x}$ .

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) (\frac{d}{d u_{3}} [\ln (u_{3})] \frac{d}{d x} [\frac{1}{x}])) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.3.2

The derivative of $\ln (u_{3})$ with respect to $u_{3}$ is $\frac{1}{u_{3}}$ .

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) (\frac{1}{u_{3}} \frac{d}{d x} [\frac{1}{x}])) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.3.3

Replace all occurrences of $u_{3}$ with $\frac{1}{x}$ .

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) (\frac{1}{\frac{1}{x}} \frac{d}{d x} [\frac{1}{x}])) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.4

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

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) (\frac{1}{\frac{1}{x}} \frac{d}{d x} [x^{- 1}])) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.5

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

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) (\frac{1}{\frac{1}{x}} (- x^{- 2}))) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.6

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

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) (1 x (- x^{- 2}))) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.7

Multiply $x$ by $1$ .

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) (x (- x^{- 2}))) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.8

Raise $x$ to the power of $1$ .

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) (- (x^{1} x^{- 2}))) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.9

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

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) (- x^{1 - 2})) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.10

Subtract $2$ from $1$ .

$2 \cos (\ln (\frac{1}{x})) (- \sin (\ln (\frac{1}{x})) (- x^{- 1})) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.11

Multiply $- 1$ by $- 1$ .

$2 \cos (\ln (\frac{1}{x})) (1 \sin (\ln (\frac{1}{x})) x^{- 1}) + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 2.12

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

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} + \frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$

Step 3

Evaluate $\frac{d}{d x} [\frac{1}{2^{(\cos (\sqrt{x})) \cdot 2}}]$ .

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Step 3.1

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

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} + \frac{d}{d x} [\frac{1}{2^{2 \cdot \cos (\sqrt{x})}}]$

Step 3.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 3.3

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

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

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 2^{2 \cos (x^{\frac{1}{2}})}$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{4}$ as $2^{2 \cos (x^{\frac{1}{2}})}$ .

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

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ is $n {u_{4}}^{n - 1}$ where $n = - 1$ .

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

Step 3.4.3

Replace all occurrences of $u_{4}$ with $2^{2 \cos (x^{\frac{1}{2}})}$ .

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

Step 3.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = 2^{x}$ and $g (x) = 2 \cos (x^{\frac{1}{2}})$ .

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Step 3.5.1

To apply the Chain Rule, set $u_{5}$ as $2 \cos (x^{\frac{1}{2}})$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - {(2^{2 \cos (x^{\frac{1}{2}})})}^{- 2} (\frac{d}{d u_{5}} [2^{u_{5}}] \frac{d}{d x} [2 \cos (x^{\frac{1}{2}})])$

Step 3.5.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{5}} [a^{u_{5}}]$ is $a^{u_{5}} \ln (a)$ where $a$ = $2$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - {(2^{2 \cos (x^{\frac{1}{2}})})}^{- 2} (2^{u_{5}} \ln (2) \frac{d}{d x} [2 \cos (x^{\frac{1}{2}})])$

Step 3.5.3

Replace all occurrences of $u_{5}$ with $2 \cos (x^{\frac{1}{2}})$ .

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

Step 3.6

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

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

Step 3.7

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 3.7.1

To apply the Chain Rule, set $u_{6}$ as $x^{\frac{1}{2}}$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - {(2^{2 \cos (x^{\frac{1}{2}})})}^{- 2} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (2 (\frac{d}{d u_{6}} [\cos (u_{6})] \frac{d}{d x} [x^{\frac{1}{2}}])))$

Step 3.7.2

The derivative of $\cos (u_{6})$ with respect to $u_{6}$ is $- \sin (u_{6})$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - {(2^{2 \cos (x^{\frac{1}{2}})})}^{- 2} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (2 (- \sin (u_{6}) \frac{d}{d x} [x^{\frac{1}{2}}])))$

Step 3.7.3

Replace all occurrences of $u_{6}$ with $x^{\frac{1}{2}}$ .

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

Step 3.8

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

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

Step 3.9

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

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Step 3.9.1

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

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

Step 3.9.2

Multiply $- 2$ by $2$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (2 (- \sin (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1}{2} - 1}))))$

Step 3.10

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (2 (- \sin (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}))))$

Step 3.11

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

Step 3.12

Combine the numerators over the common denominator.

Step 3.13

Simplify the numerator.

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Step 3.13.1

Multiply $- 1$ by $2$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (2 (- \sin (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{1 - 2}{2}}))))$

Step 3.13.2

Subtract $2$ from $1$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (2 (- \sin (x^{\frac{1}{2}}) (\frac{1}{2} x^{\frac{- 1}{2}}))))$

Step 3.14

Move the negative in front of the fraction.

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (2 (- \sin (x^{\frac{1}{2}}) (\frac{1}{2} x^{- \frac{1}{2}}))))$

Step 3.15

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

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (2 (- \sin (x^{\frac{1}{2}}) \frac{x^{- \frac{1}{2}}}{2})))$

Step 3.16

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

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (2 (- \frac{x^{- \frac{1}{2}} \sin (x^{\frac{1}{2}})}{2})))$

Step 3.17

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (2 (- \frac{\sin (x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}})))$

Step 3.18

Multiply $- 1$ by $2$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (- 2 \frac{\sin (x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}}))$

Step 3.19

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

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) \frac{- 2 \sin (x^{\frac{1}{2}})}{2 x^{\frac{1}{2}}})$

Step 3.20

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

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) \frac{2 (- \sin (x^{\frac{1}{2}}))}{2 x^{\frac{1}{2}}})$

Step 3.21

Cancel the common factors.

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Step 3.21.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) \frac{2 (- \sin (x^{\frac{1}{2}}))}{2 (x^{\frac{1}{2}})})$

Step 3.21.2

Cancel the common factor.

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) \frac{2 (- \sin (x^{\frac{1}{2}}))}{2 x^{\frac{1}{2}}})$

Step 3.21.3

Rewrite the expression.

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) \frac{- \sin (x^{\frac{1}{2}})}{x^{\frac{1}{2}}})$

Step 3.22

Move the negative in front of the fraction.

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (2^{2 \cos (x^{\frac{1}{2}})} \ln (2) (- \frac{\sin (x^{\frac{1}{2}})}{x^{\frac{1}{2}}}))$

Step 3.23

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

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (\ln (2) (- \frac{2^{2 \cos (x^{\frac{1}{2}})} \sin (x^{\frac{1}{2}})}{x^{\frac{1}{2}}}))$

Step 3.24

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

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} - 2^{- 4 \cos (x^{\frac{1}{2}})} (- \frac{\ln (2) (2^{2 \cos (x^{\frac{1}{2}})} \sin (x^{\frac{1}{2}}))}{x^{\frac{1}{2}}})$

Step 3.25

Multiply $- 1$ by $- 1$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} + 1 \cdot 2^{- 4 \cos (x^{\frac{1}{2}})} \frac{\ln (2) \cdot 2^{2 \cos (x^{\frac{1}{2}})} \sin (x^{\frac{1}{2}})}{x^{\frac{1}{2}}}$

Step 3.26

Multiply $2^{- 4 \cos (x^{\frac{1}{2}})}$ by $1$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} + 2^{- 4 \cos (x^{\frac{1}{2}})} \frac{\ln (2) \cdot 2^{2 \cos (x^{\frac{1}{2}})} \sin (x^{\frac{1}{2}})}{x^{\frac{1}{2}}}$

Step 3.27

Combine $2^{- 4 \cos (x^{\frac{1}{2}})}$ and $\frac{\ln (2) \cdot 2^{2 \cos (x^{\frac{1}{2}})} \sin (x^{\frac{1}{2}})}{x^{\frac{1}{2}}}$ .

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} + \frac{2^{- 4 \cos (x^{\frac{1}{2}})} (\ln (2) \cdot 2^{2 \cos (x^{\frac{1}{2}})} \sin (x^{\frac{1}{2}}))}{x^{\frac{1}{2}}}$

Step 3.28

Multiply $2^{- 4 \cos (x^{\frac{1}{2}})}$ by $2^{2 \cos (x^{\frac{1}{2}})}$ by adding the exponents.

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Step 3.28.1

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

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} + \frac{2^{2 \cos (x^{\frac{1}{2}})} \cdot 2^{- 4 \cos (x^{\frac{1}{2}})} (\ln (2) \sin (x^{\frac{1}{2}}))}{x^{\frac{1}{2}}}$

Step 3.28.2

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

$2 \cos (\ln (\frac{1}{x})) \sin (\ln (\frac{1}{x})) x^{- 1} + \frac{2^{2 \cos (x^{\frac{1}{2}}) - 4 \cos (x^{\frac{1}{2}})} (\ln (2) \sin (x^{\frac{1}{2}}))}{x^{\frac{1}{2}}}$

Step 3.28.3

Subtract $4 \cos (x^{\frac{1}{2}})$ from $2 \cos (x^{\frac{1}{2}})$ .

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Combine terms.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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