Calculus Examples

$f (x) = \frac{\cot (x)}{e^{x}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \cot (x)$ and $g (x) = e^{x}$ .

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

Step 2

Multiply the exponents in ${(e^{x})}^{2}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2

Move $2$ to the left of $x$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.1.2

Reorder factors in $- e^{x} \csc^{2} (x) - \cot (x) e^{x}$ .

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

Step 5.2

Factor $e^{x}$ out of $- e^{x} \csc^{2} (x) - e^{x} \cot (x)$ .

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Step 5.2.1

Factor $e^{x}$ out of $- e^{x} \csc^{2} (x)$ .

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

Step 5.2.2

Factor $e^{x}$ out of $- e^{x} \cot (x)$ .

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

Step 5.2.3

Factor $e^{x}$ out of $e^{x} (- \csc^{2} (x)) + e^{x} (- \cot (x))$ .

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

Step 5.3

Cancel the common factor of $e^{x}$ and $e^{2 x}$ .

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Step 5.3.1

Factor $e^{2 x}$ out of $e^{x} (- \csc^{2} (x) - \cot (x))$ .

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

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Multiply by $1$ .

$\frac{e^{2 x} (e^{- x} (- \csc^{2} (x) - \cot (x)))}{e^{2 x} \cdot 1}$

Step 5.3.2.2

Cancel the common factor.

$\frac{e^{2 x} (e^{- x} (- \csc^{2} (x) - \cot (x)))}{e^{2 x} \cdot 1}$

Step 5.3.2.3

Rewrite the expression.

$\frac{e^{- x} (- \csc^{2} (x) - \cot (x))}{1}$

Step 5.3.2.4

Divide $e^{- x} (- \csc^{2} (x) - \cot (x))$ by $1$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Rewrite using the commutative property of multiplication.

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

Step 5.6

Rewrite using the commutative property of multiplication.

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