Calculus Examples

$f (x) = 5 e^{x} \cos (x)$

Step 1

Find the first derivative.

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Step 1.1

Since $5$ is constant with respect to $x$ , the derivative of $5 e^{x} \cos (x)$ with respect to $x$ is $5 \frac{d}{d x} [e^{x} \cos (x)]$ .

$5 \frac{d}{d x} [e^{x} \cos (x)]$

Step 1.2

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 $f (x) = e^{x}$ and $g (x) = \cos (x)$ .

$5 (e^{x} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [e^{x}])$

Step 1.3

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

$5 (e^{x} (- \sin (x)) + \cos (x) \frac{d}{d x} [e^{x}])$

Step 1.4

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

$5 (e^{x} (- \sin (x)) + \cos (x) e^{x})$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$5 (e^{x} (- \sin (x))) + 5 (\cos (x) e^{x})$

Step 1.5.2

Multiply $- 1$ by $5$ .

$- 5 e^{x} \sin (x) + 5 \cos (x) e^{x}$

Step 1.5.3

Reorder terms.

$f' (x) = - 5 e^{x} \sin (x) + 5 e^{x} \cos (x)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 5 e^{x} \sin (x) + 5 e^{x} \cos (x)$ with respect to $x$ is $\frac{d}{d x} [- 5 e^{x} \sin (x)] + \frac{d}{d x} [5 e^{x} \cos (x)]$ .

$\frac{d}{d x} [- 5 e^{x} \sin (x)] + \frac{d}{d x} [5 e^{x} \cos (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [- 5 e^{x} \sin (x)]$ .

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Step 2.2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 e^{x} \sin (x)$ with respect to $x$ is $- 5 \frac{d}{d x} [e^{x} \sin (x)]$ .

$- 5 \frac{d}{d x} [e^{x} \sin (x)] + \frac{d}{d x} [5 e^{x} \cos (x)]$

Step 2.2.2

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 $f (x) = e^{x}$ and $g (x) = \sin (x)$ .

$- 5 (e^{x} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [e^{x}]) + \frac{d}{d x} [5 e^{x} \cos (x)]$

Step 2.2.3

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

$- 5 (e^{x} \cos (x) + \sin (x) \frac{d}{d x} [e^{x}]) + \frac{d}{d x} [5 e^{x} \cos (x)]$

Step 2.2.4

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

$- 5 (e^{x} \cos (x) + \sin (x) e^{x}) + \frac{d}{d x} [5 e^{x} \cos (x)]$

Step 2.3

Evaluate $\frac{d}{d x} [5 e^{x} \cos (x)]$ .

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Step 2.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 e^{x} \cos (x)$ with respect to $x$ is $5 \frac{d}{d x} [e^{x} \cos (x)]$ .

$- 5 (e^{x} \cos (x) + \sin (x) e^{x}) + 5 \frac{d}{d x} [e^{x} \cos (x)]$

Step 2.3.2

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 $f (x) = e^{x}$ and $g (x) = \cos (x)$ .

$- 5 (e^{x} \cos (x) + \sin (x) e^{x}) + 5 (e^{x} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [e^{x}])$

Step 2.3.3

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

$- 5 (e^{x} \cos (x) + \sin (x) e^{x}) + 5 (e^{x} (- \sin (x)) + \cos (x) \frac{d}{d x} [e^{x}])$

Step 2.3.4

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

$- 5 (e^{x} \cos (x) + \sin (x) e^{x}) + 5 (e^{x} (- \sin (x)) + \cos (x) e^{x})$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$- 5 (e^{x} \cos (x)) - 5 (\sin (x) e^{x}) + 5 (e^{x} (- \sin (x)) + \cos (x) e^{x})$

Step 2.4.2

Apply the distributive property.

$- 5 (e^{x} \cos (x)) - 5 (\sin (x) e^{x}) + 5 (e^{x} (- \sin (x))) + 5 (\cos (x) e^{x})$

Step 2.4.3

Combine terms.

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Step 2.4.3.1

Multiply $- 1$ by $5$ .

$- 5 e^{x} \cos (x) - 5 \sin (x) e^{x} - 5 (e^{x} (\sin (x))) + 5 (\cos (x) e^{x})$

Step 2.4.3.2

Subtract $5 e^{x} \sin (x)$ from $- 5 \sin (x) e^{x}$ .

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Step 2.4.3.2.1

Move $\sin (x)$ .

$- 5 e^{x} \cos (x) - 5 e^{x} \sin (x) - 5 e^{x} \sin (x) + 5 \cos (x) e^{x}$

Step 2.4.3.2.2

Subtract $5 e^{x} \sin (x)$ from $- 5 e^{x} \sin (x)$ .

$- 5 e^{x} \cos (x) - 10 e^{x} \sin (x) + 5 \cos (x) e^{x}$

Step 2.4.3.3

Add $- 5 e^{x} \cos (x)$ and $5 \cos (x) e^{x}$ .

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Step 2.4.3.3.1

Move $\cos (x)$ .

$- 10 e^{x} \sin (x) - 5 e^{x} \cos (x) + 5 e^{x} \cos (x)$

Step 2.4.3.3.2

Add $- 5 e^{x} \cos (x)$ and $5 e^{x} \cos (x)$ .

$- 10 e^{x} \sin (x) + 0$

Step 2.4.3.4

Add $- 10 e^{x} \sin (x)$ and $0$ .

$f'' (x) = - 10 e^{x} \sin (x)$

Step 3

Find the third derivative.

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Step 3.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 e^{x} \sin (x)$ with respect to $x$ is $- 10 \frac{d}{d x} [e^{x} \sin (x)]$ .

$- 10 \frac{d}{d x} [e^{x} \sin (x)]$

Step 3.2

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 $f (x) = e^{x}$ and $g (x) = \sin (x)$ .

$- 10 (e^{x} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [e^{x}])$

Step 3.3

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

$- 10 (e^{x} \cos (x) + \sin (x) \frac{d}{d x} [e^{x}])$

Step 3.4

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

$- 10 (e^{x} \cos (x) + \sin (x) e^{x})$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$- 10 (e^{x} \cos (x)) - 10 (\sin (x) e^{x})$

Step 3.5.2

Remove parentheses.

$- 10 e^{x} \cos (x) - 10 \sin (x) e^{x}$

Step 3.5.3

Reorder terms.

$f''' (x) = - 10 e^{x} \cos (x) - 10 e^{x} \sin (x)$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $- 10 e^{x} \cos (x) - 10 e^{x} \sin (x)$ .

$- 10 e^{x} \cos (x) - 10 e^{x} \sin (x)$