Calculus Examples

$y = {\cos (2 x)}^{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({\cos (2 x)}^{x})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Use the properties of logarithms to simplify the differentiation.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

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 $g (x) = \ln (\cos (2 x))$ .

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

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

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

Step 3.6

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

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Step 3.6.1

To apply the Chain Rule, set $u_{3}$ as $2 x$ .

$e^{x \ln (\cos (2 x))} (x (\sec (2 x) (\frac{d}{d u_{3}} [\cos (u_{3})] \frac{d}{d x} [2 x])) + \ln (\cos (2 x)) \frac{d}{d x} [x])$

Step 3.6.2

The derivative of $\cos (u_{3})$ with respect to $u_{3}$ is $- \sin (u_{3})$ .

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

Step 3.6.3

Replace all occurrences of $u_{3}$ with $2 x$ .

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

Step 3.7

Differentiate.

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Step 3.7.1

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

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

Step 3.7.2

Multiply $2$ by $- 1$ .

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

Step 3.7.3

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

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

Step 3.7.4

Multiply $- 2$ by $1$ .

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

Step 3.7.5

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

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

Step 3.7.6

Multiply $\ln (\cos (2 x))$ by $1$ .

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Remove parentheses.

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

Step 3.8.3

Reorder terms.

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

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - 2 e^{x \ln (\cos (2 x))} x \sec (2 x) \sin (2 x) + e^{x \ln (\cos (2 x))} \ln (\cos (2 x))$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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