Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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 $g (x) = \sin (x)$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

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

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Step 2.3.1

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 $g (x) = \sin (x)$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Combine terms.

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Step 2.4.1

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

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Step 2.4.1.1

Reorder $\cos (x)$ and $e^{x}$ .

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

Step 2.4.1.2

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

$e^{x} (- \sin (x)) + 2 e^{x} \cos (x) + \sin (x) e^{x}$

Step 2.4.2

Reorder $\sin (x)$ and $e^{x}$ .

$2 e^{x} \cos (x) - 1 \cdot e^{x} \sin (x) + e^{x} \sin (x)$

Step 2.4.3

Rewrite $- 1 e^{x}$ as $- e^{x}$ .

$2 e^{x} \cos (x) - e^{x} \sin (x) + e^{x} \sin (x)$

Step 2.4.4

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

$2 e^{x} \cos (x) + 0$

Step 2.4.5

Add $2 e^{x} \cos (x)$ and $0$ .

$f'' (x) = 2 e^{x} \cos (x)$

Step 3

Find the third derivative.

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Step 3.1

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

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

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

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

Step 3.3

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

$2 (e^{x} (- \sin (x)) + \cos (x) \frac{d}{d x} [e^{x}])$

Step 3.4

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

$2 (e^{x} (- \sin (x)) + \cos (x) e^{x})$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$2 (e^{x} (- \sin (x))) + 2 (\cos (x) e^{x})$

Step 3.5.2

Multiply $- 1$ by $2$ .

$- 2 e^{x} \sin (x) + 2 \cos (x) e^{x}$

Step 3.5.3

Reorder terms.

$f''' (x) = - 2 e^{x} \sin (x) + 2 e^{x} \cos (x)$

Step 4

The third derivative of $f (x)$ with respect to $x$ is $- 2 e^{x} \sin (x) + 2 e^{x} \cos (x)$ .

$- 2 e^{x} \sin (x) + 2 e^{x} \cos (x)$