Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

Step 3.1.2

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

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

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

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

Step 3.7.2

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

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

Step 3.7.3

Reorder terms.

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

Step 4

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

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

Step 5

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

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