Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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

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

Step 3.1.2

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

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

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

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Step 3.2.1

To apply the Chain Rule, set $u$ as $2 \cot (x) \ln (x)$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $2 \cot (x) \ln (x)$ .

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

Step 3.3

Differentiate using the Constant Multiple Rule.

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Step 3.3.1

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

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

Step 3.3.2

Move $2$ to the left of $e^{2 \cot (x) \ln (x)}$ .

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

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$2 e^{2 \cot (x) \ln (x)} (\frac{\cot (x)}{x} + \ln (x) (- \csc^{2} (x)))$

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

$2 e^{2 \cot (x) \ln (x)} \frac{\cot (x)}{x} + 2 e^{2 \cot (x) \ln (x)} (\ln (x) (- \csc^{2} (x)))$

Step 3.8.2

Combine terms.

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Step 3.8.2.1

Combine $\frac{\cot (x)}{x}$ and $2$ .

$\frac{\cot (x) \cdot 2}{x} e^{2 \cot (x) \ln (x)} + 2 e^{2 \cot (x) \ln (x)} (\ln (x) (- \csc^{2} (x)))$

Step 3.8.2.2

Combine $\frac{\cot (x) \cdot 2}{x}$ and $e^{2 \cot (x) \ln (x)}$ .

$\frac{\cot (x) \cdot 2 e^{2 \cot (x) \ln (x)}}{x} + 2 e^{2 \cot (x) \ln (x)} (\ln (x) (- \csc^{2} (x)))$

Step 3.8.2.3

Move $2$ to the left of $\cot (x)$ .

$\frac{2 \cdot \cot (x) e^{2 \cot (x) \ln (x)}}{x} + 2 e^{2 \cot (x) \ln (x)} (\ln (x) (- \csc^{2} (x)))$

Step 3.8.2.4

Multiply $- 1$ by $2$ .

$\frac{2 \cot (x) e^{2 \cot (x) \ln (x)}}{x} - 2 e^{2 \cot (x) \ln (x)} \ln (x) \csc^{2} (x)$

Step 3.8.3

Reorder terms.

$- 2 e^{2 \cot (x) \ln (x)} \csc^{2} (x) \ln (x) + \frac{2 \cot (x) e^{2 \cot (x) \ln (x)}}{x}$

Step 4

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

$y' = - 2 e^{2 \cot (x) \ln (x)} \csc^{2} (x) \ln (x) + \frac{2 \cot (x) e^{2 \cot (x) \ln (x)}}{x}$

Step 5

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

$\frac{d y}{d x} = - 2 e^{2 \cot (x) \ln (x)} \csc^{2} (x) \ln (x) + \frac{2 \cot (x) e^{2 \cot (x) \ln (x)}}{x}$