Calculus Examples

$y = {\cos (6 x)}^{x}$

Step 1

Let $y = f (x)$ , take the natural logarithm of both sides $\ln (y) = \ln (f (x))$ .

$\ln (y) = \ln ({\cos (6 x)}^{x})$

Step 2

Expand $\ln ({\cos (6 x)}^{x})$ by moving $x$ outside the logarithm.

$\ln (y) = x \ln (\cos (6 x))$

Step 3

Differentiate the expression using the chain rule, keeping in mind that $y$ is a function of $x$ .

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Step 3.1

Differentiate the left hand side $\ln (y)$ using the chain rule.

$\frac{y'}{y} = x \ln (\cos (6 x))$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $x \ln (\cos (6 x))$ .

$\frac{y'}{y} = \frac{d}{d x} [x \ln (\cos (6 x))]$

Step 3.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (\cos (6 x))$ .

$\frac{y'}{y} = x \frac{d}{d x} [\ln (\cos (6 x))] + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.3

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 $g (x) = \cos (6 x)$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $\cos (6 x)$ .

$\frac{y'}{y} = x (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [\cos (6 x)]) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.3.2

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

$\frac{y'}{y} = x (\frac{1}{u_{1}} \frac{d}{d x} [\cos (6 x)]) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $\cos (6 x)$ .

$\frac{y'}{y} = x (\frac{1}{\cos (6 x)} \frac{d}{d x} [\cos (6 x)]) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.4

Convert from $\frac{1}{\cos (6 x)}$ to $\sec (6 x)$ .

$\frac{y'}{y} = x (\sec (6 x) \frac{d}{d x} [\cos (6 x)]) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 6 x$ .

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Step 3.2.5.1

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

$\frac{y'}{y} = x (\sec (6 x) (\frac{d}{d u_{2}} [\cos (u_{2})] \frac{d}{d x} [6 x])) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.5.2

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

$\frac{y'}{y} = x (\sec (6 x) (- \sin (u_{2}) \frac{d}{d x} [6 x])) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.5.3

Replace all occurrences of $u_{2}$ with $6 x$ .

$\frac{y'}{y} = x (\sec (6 x) (- \sin (6 x) \frac{d}{d x} [6 x])) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.6

Differentiate.

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Step 3.2.6.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\frac{y'}{y} = x (\sec (6 x) (- \sin (6 x) (6 \frac{d}{d x} [x]))) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.6.2

Multiply $6$ by $- 1$ .

$\frac{y'}{y} = x (\sec (6 x) (- 6 \sin (6 x) \frac{d}{d x} [x])) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.6.3

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

$\frac{y'}{y} = x (\sec (6 x) (- 6 \sin (6 x) \cdot 1)) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.6.4

Multiply $- 6$ by $1$ .

$\frac{y'}{y} = x (\sec (6 x) (- 6 \sin (6 x))) + \ln (\cos (6 x)) \frac{d}{d x} [x]$

Step 3.2.6.5

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

$\frac{y'}{y} = x (\sec (6 x) (- 6 \sin (6 x))) + \ln (\cos (6 x)) \cdot 1$

Step 3.2.6.6

Multiply $\ln (\cos (6 x))$ by $1$ .

$\frac{y'}{y} = x (\sec (6 x) (- 6 \sin (6 x))) + \ln (\cos (6 x))$

Step 3.2.7

Simplify.

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Step 3.2.7.1

Reorder terms.

$\frac{y'}{y} = - 6 x \sec (6 x) \sin (6 x) + \ln (\cos (6 x))$

Step 3.2.7.2

Simplify each term.

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Step 3.2.7.2.1

Rewrite $\sec (6 x)$ in terms of sines and cosines.

$\frac{y'}{y} = - 6 x \frac{1}{\cos (6 x)} \sin (6 x) + \ln (\cos (6 x))$

Step 3.2.7.2.2

Multiply $- 6 x \frac{1}{\cos (6 x)}$ .

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Step 3.2.7.2.2.1

Combine $- 6$ and $\frac{1}{\cos (6 x)}$ .

$\frac{y'}{y} = x \frac{- 6}{\cos (6 x)} \sin (6 x) + \ln (\cos (6 x))$

Step 3.2.7.2.2.2

Combine $x$ and $\frac{- 6}{\cos (6 x)}$ .

$\frac{y'}{y} = \frac{x \cdot - 6}{\cos (6 x)} \sin (6 x) + \ln (\cos (6 x))$

Step 3.2.7.2.3

Move $- 6$ to the left of $x$ .

$\frac{y'}{y} = \frac{- 6 \cdot x}{\cos (6 x)} \sin (6 x) + \ln (\cos (6 x))$

Step 3.2.7.2.4

Move the negative in front of the fraction.

$\frac{y'}{y} = - \frac{6 \cdot x}{\cos (6 x)} \sin (6 x) + \ln (\cos (6 x))$

Step 3.2.7.2.5

Combine $\sin (6 x)$ and $\frac{6 x}{\cos (6 x)}$ .

$\frac{y'}{y} = - \frac{\sin (6 x) (6 x)}{\cos (6 x)} + \ln (\cos (6 x))$

Step 3.2.7.2.6

Move $6$ to the left of $\sin (6 x)$ .

$\frac{y'}{y} = - \frac{6 \sin (6 x) x}{\cos (6 x)} + \ln (\cos (6 x))$

Step 3.2.7.3

Simplify each term.

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Step 3.2.7.3.1

Separate fractions.

$\frac{y'}{y} = - (\frac{6 x}{1} \cdot \frac{\sin (6 x)}{\cos (6 x)}) + \ln (\cos (6 x))$

Step 3.2.7.3.2

Convert from $\frac{\sin (6 x)}{\cos (6 x)}$ to $\tan (6 x)$ .

$\frac{y'}{y} = - (\frac{6 x}{1} \tan (6 x)) + \ln (\cos (6 x))$

Step 3.2.7.3.3

Divide $6 x$ by $1$ .

$\frac{y'}{y} = - (6 x \tan (6 x)) + \ln (\cos (6 x))$

Step 3.2.7.3.4

Multiply $6$ by $- 1$ .

$\frac{y'}{y} = - 6 x \tan (6 x) + \ln (\cos (6 x))$

Step 4

Isolate $y'$ and substitute the original function for $y$ in the right hand side.

$y' = (- 6 x \tan (6 x) + \ln (\cos (6 x))) {\cos (6 x)}^{x}$

Step 5

Simplify the right hand side.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite $\tan (6 x)$ in terms of sines and cosines.

$y' = (- 6 x \frac{\sin (6 x)}{\cos (6 x)} + \ln (\cos (6 x))) {\cos (6 x)}^{x}$

Step 5.1.2

Multiply $- 6 x \frac{\sin (6 x)}{\cos (6 x)}$ .

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Step 5.1.2.1

Combine $- 6$ and $\frac{\sin (6 x)}{\cos (6 x)}$ .

$y' = (x \frac{- 6 \sin (6 x)}{\cos (6 x)} + \ln (\cos (6 x))) {\cos (6 x)}^{x}$

Step 5.1.2.2

Combine $x$ and $\frac{- 6 \sin (6 x)}{\cos (6 x)}$ .

$y' = (\frac{x (- 6 \sin (6 x))}{\cos (6 x)} + \ln (\cos (6 x))) {\cos (6 x)}^{x}$

Step 5.1.3

Move $- 6$ to the left of $x$ .

$y' = (\frac{- 6 \cdot x \sin (6 x)}{\cos (6 x)} + \ln (\cos (6 x))) {\cos (6 x)}^{x}$

Step 5.1.4

Move the negative in front of the fraction.

$y' = (- \frac{6 x \sin (6 x)}{\cos (6 x)} + \ln (\cos (6 x))) {\cos (6 x)}^{x}$

Step 5.2

Apply the distributive property.

$y' = - \frac{6 x \sin (6 x)}{\cos (6 x)} {\cos (6 x)}^{x} + \ln (\cos (6 x)) {\cos (6 x)}^{x}$

Step 5.3

Combine ${\cos (6 x)}^{x}$ and $\frac{6 x \sin (6 x)}{\cos (6 x)}$ .

$y' = - \frac{{\cos (6 x)}^{x} (6 x \sin (6 x))}{\cos (6 x)} + \ln (\cos (6 x)) {\cos (6 x)}^{x}$

Step 5.4

Simplify each term.

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Step 5.4.1

Cancel the common factor of ${\cos (6 x)}^{x}$ and $\cos (6 x)$ .

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Step 5.4.1.1

Factor $\cos (6 x)$ out of ${\cos (6 x)}^{x} (6 x \sin (6 x))$ .

$y' = - \frac{\cos (6 x) ({\cos (6 x)}^{x - 1} (6 x \sin (6 x)))}{\cos (6 x)} + \ln (\cos (6 x)) {\cos (6 x)}^{x}$

Step 5.4.1.2

Cancel the common factors.

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Step 5.4.1.2.1

Multiply by $1$ .

$y' = - \frac{\cos (6 x) ({\cos (6 x)}^{x - 1} (6 x \sin (6 x)))}{\cos (6 x) \cdot 1} + \ln (\cos (6 x)) {\cos (6 x)}^{x}$

Step 5.4.1.2.2

Cancel the common factor.

$y' = - \frac{\cos (6 x) ({\cos (6 x)}^{x - 1} (6 x \sin (6 x)))}{\cos (6 x) \cdot 1} + \ln (\cos (6 x)) {\cos (6 x)}^{x}$

Step 5.4.1.2.3

Rewrite the expression.

$y' = - \frac{{\cos (6 x)}^{x - 1} (6 x \sin (6 x))}{1} + \ln (\cos (6 x)) {\cos (6 x)}^{x}$

Step 5.4.1.2.4

Divide ${\cos (6 x)}^{x - 1} (6 x \sin (6 x))$ by $1$ .

$y' = - ({\cos (6 x)}^{x - 1} (6 x \sin (6 x))) + \ln (\cos (6 x)) {\cos (6 x)}^{x}$

Step 5.4.2

Rewrite using the commutative property of multiplication.

$y' = - (6 {\cos (6 x)}^{x - 1} x \sin (6 x)) + \ln (\cos (6 x)) {\cos (6 x)}^{x}$

Step 5.4.3

Multiply $6$ by $- 1$ .

$y' = - 6 {\cos (6 x)}^{x - 1} x \sin (6 x) + \ln (\cos (6 x)) {\cos (6 x)}^{x}$

Step 5.5

Reorder factors in $- 6 {\cos (6 x)}^{x - 1} x \sin (6 x) + \ln (\cos (6 x)) {\cos (6 x)}^{x}$ .

$y' = - 6 x {\cos (6 x)}^{x - 1} \sin (6 x) + {\cos (6 x)}^{x} \ln (\cos (6 x))$