Calculus Examples

$y = x^{- 4 \cos (x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{- 4 \cos (x)})$

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 $x^{- 4 \cos (x)}$ as $e^{\ln (x^{- 4 \cos (x)})}$ .

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

Step 3.1.2

Expand $\ln (x^{- 4 \cos (x)})$ by moving $- 4 \cos (x)$ outside the logarithm.

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

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 $g (x) = - 4 \cos (x) \ln (x)$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $- 4 \cos (x) \ln (x)$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $- 4 \cos (x) \ln (x)$ .

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

Step 3.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \cos (x) \ln (x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\cos (x) \ln (x)]$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Apply the distributive property.

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

Step 3.8.3

Combine terms.

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Step 3.8.3.1

Combine $- 4$ and $\frac{\cos (x)}{x}$ .

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

Step 3.8.3.2

Move the negative in front of the fraction.

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

Step 3.8.3.3

Combine $e^{- 4 \cos (x) \ln (x)}$ and $\frac{4 \cos (x)}{x}$ .

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

Step 3.8.3.4

Move $4$ to the left of $e^{- 4 \cos (x) \ln (x)}$ .

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

Step 3.8.3.5

Multiply $- 1$ by $- 4$ .

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

Step 3.8.4

Reorder terms.

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

Step 4

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

$y' = 4 e^{- 4 \cos (x) \ln (x)} \sin (x) \ln (x) - \frac{4 e^{- 4 \cos (x) \ln (x)} \cos (x)}{x}$

Step 5

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

$\frac{d y}{d x} = 4 e^{- 4 \cos (x) \ln (x)} \sin (x) \ln (x) - \frac{4 e^{- 4 \cos (x) \ln (x)} \cos (x)}{x}$