Calculus Examples

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

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

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Step 1.1

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

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

Step 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 $n = 2$ .

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

Step 1.3

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

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

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Multiply $- 1$ by $2$ .

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

Step 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 $g (x) = 2 x$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

Simplify terms.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

Cancel the common factor of $- 2$ and $2$ .

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Step 5.2.3.1

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

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

Step 5.2.3.2

Cancel the common factors.

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Step 5.2.3.2.1

Factor $2$ out of $2 x$ .

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

Step 5.2.3.2.2

Cancel the common factor.

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

Step 5.2.3.2.3

Rewrite the expression.

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

Step 5.2.4

Move the negative in front of the fraction.

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

Step 5.3

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

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

Step 5.4

Combine fractions.

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Step 5.4.1

Multiply $2$ by $- 1$ .

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

Step 5.4.2

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

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

Step 5.4.3

Move the negative in front of the fraction.

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

Step 5.5

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

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

Step 5.6

Multiply $- 1$ by $1$ .

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

Step 6

Simplify the numerator.

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Step 6.1

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

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

Step 6.2

Reorder $\sin (\ln (2 x))$ and $2$ .

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

Step 6.3

Apply the sine double-angle identity.

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

Step 6.4

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

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

Step 6.5

Apply the product rule to $2 x$ .

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

Step 6.6

Raise $2$ to the power of $2$ .

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