Calculus Examples

$f (x) = e^{x} \sin (x) - 3 e^{x} \cos (x)$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $e^{x} \sin (x) - 3 e^{x} \cos (x)$ with respect to $x$ is $\frac{d}{d x} [e^{x} \sin (x)] + \frac{d}{d x} [- 3 e^{x} \cos (x)]$ .

$\frac{d}{d x} [e^{x} \sin (x)] + \frac{d}{d x} [- 3 e^{x} \cos (x)]$

Step 1.2

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

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Step 1.2.1

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

$e^{x} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 3 e^{x} \cos (x)]$

Step 1.2.2

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

$e^{x} \cos (x) + \sin (x) \frac{d}{d x} [e^{x}] + \frac{d}{d x} [- 3 e^{x} \cos (x)]$

Step 1.2.3

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

$e^{x} \cos (x) + \sin (x) e^{x} + \frac{d}{d x} [- 3 e^{x} \cos (x)]$

Step 1.3

Evaluate $\frac{d}{d x} [- 3 e^{x} \cos (x)]$ .

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Step 1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 e^{x} \cos (x)$ with respect to $x$ is $- 3 \frac{d}{d x} [e^{x} \cos (x)]$ .

$e^{x} \cos (x) + \sin (x) e^{x} - 3 \frac{d}{d x} [e^{x} \cos (x)]$

Step 1.3.2

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

$e^{x} \cos (x) + \sin (x) e^{x} - 3 (e^{x} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [e^{x}])$

Step 1.3.3

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

$e^{x} \cos (x) + \sin (x) e^{x} - 3 (e^{x} (- \sin (x)) + \cos (x) \frac{d}{d x} [e^{x}])$

Step 1.3.4

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

$e^{x} \cos (x) + \sin (x) e^{x} - 3 (e^{x} (- \sin (x)) + \cos (x) e^{x})$

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

$e^{x} \cos (x) + \sin (x) e^{x} - 3 (e^{x} (- \sin (x))) - 3 (\cos (x) e^{x})$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Multiply $- 1$ by $- 3$ .

$e^{x} \cos (x) + \sin (x) e^{x} + 3 (e^{x} (\sin (x))) - 3 (\cos (x) e^{x})$

Step 1.4.2.2

Add $\sin (x) e^{x}$ and $3 e^{x} \sin (x)$ .

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Step 1.4.2.2.1

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

$e^{x} \cos (x) + e^{x} \sin (x) + 3 e^{x} \sin (x) - 3 \cos (x) e^{x}$

Step 1.4.2.2.2

Add $e^{x} \sin (x)$ and $3 e^{x} \sin (x)$ .

$e^{x} \cos (x) + 4 e^{x} \sin (x) - 3 \cos (x) e^{x}$

Step 1.4.2.3

Subtract $3 \cos (x) e^{x}$ from $e^{x} \cos (x)$ .

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Step 1.4.2.3.1

Move $\cos (x)$ .

$e^{x} \cos (x) - 3 e^{x} \cos (x) + 4 e^{x} \sin (x)$

Step 1.4.2.3.2

Subtract $3 e^{x} \cos (x)$ from $e^{x} \cos (x)$ .

$f' (x) = - 2 e^{x} \cos (x) + 4 e^{x} \sin (x)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 2 e^{x} \cos (x) + 4 e^{x} \sin (x)$ with respect to $x$ is $\frac{d}{d x} [- 2 e^{x} \cos (x)] + \frac{d}{d x} [4 e^{x} \sin (x)]$ .

$\frac{d}{d x} [- 2 e^{x} \cos (x)] + \frac{d}{d x} [4 e^{x} \sin (x)]$

Step 2.2

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

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Step 2.2.1

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

$- 2 \frac{d}{d x} [e^{x} \cos (x)] + \frac{d}{d x} [4 e^{x} \sin (x)]$

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

$- 2 (e^{x} \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [e^{x}]) + \frac{d}{d x} [4 e^{x} \sin (x)]$

Step 2.2.3

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

$- 2 (e^{x} (- \sin (x)) + \cos (x) \frac{d}{d x} [e^{x}]) + \frac{d}{d x} [4 e^{x} \sin (x)]$

Step 2.2.4

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

$- 2 (e^{x} (- \sin (x)) + \cos (x) e^{x}) + \frac{d}{d x} [4 e^{x} \sin (x)]$

Step 2.3

Evaluate $\frac{d}{d x} [4 e^{x} \sin (x)]$ .

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Step 2.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 e^{x} \sin (x)$ with respect to $x$ is $4 \frac{d}{d x} [e^{x} \sin (x)]$ .

$- 2 (e^{x} (- \sin (x)) + \cos (x) e^{x}) + 4 \frac{d}{d x} [e^{x} \sin (x)]$

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

$- 2 (e^{x} (- \sin (x)) + \cos (x) e^{x}) + 4 (e^{x} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [e^{x}])$

Step 2.3.3

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

$- 2 (e^{x} (- \sin (x)) + \cos (x) e^{x}) + 4 (e^{x} \cos (x) + \sin (x) \frac{d}{d x} [e^{x}])$

Step 2.3.4

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

$- 2 (e^{x} (- \sin (x)) + \cos (x) e^{x}) + 4 (e^{x} \cos (x) + \sin (x) e^{x})$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$- 2 (e^{x} (- \sin (x))) - 2 (\cos (x) e^{x}) + 4 (e^{x} \cos (x) + \sin (x) e^{x})$

Step 2.4.2

Apply the distributive property.

$- 2 (e^{x} (- \sin (x))) - 2 (\cos (x) e^{x}) + 4 (e^{x} \cos (x)) + 4 (\sin (x) e^{x})$

Step 2.4.3

Combine terms.

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Step 2.4.3.1

Multiply $- 1$ by $- 2$ .

$2 (e^{x} (\sin (x))) - 2 (\cos (x) e^{x}) + 4 (e^{x} \cos (x)) + 4 (\sin (x) e^{x})$

Step 2.4.3.2

Add $- 2 \cos (x) e^{x}$ and $4 e^{x} \cos (x)$ .

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Step 2.4.3.2.1

Move $\cos (x)$ .

$2 e^{x} \sin (x) - 2 e^{x} \cos (x) + 4 e^{x} \cos (x) + 4 \sin (x) e^{x}$

Step 2.4.3.2.2

Add $- 2 e^{x} \cos (x)$ and $4 e^{x} \cos (x)$ .

$2 e^{x} \sin (x) + 2 e^{x} \cos (x) + 4 \sin (x) e^{x}$

Step 2.4.3.3

Add $2 e^{x} \sin (x)$ and $4 \sin (x) e^{x}$ .

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Step 2.4.3.3.1

Move $\sin (x)$ .

$2 e^{x} \sin (x) + 4 e^{x} \sin (x) + 2 e^{x} \cos (x)$

Step 2.4.3.3.2

Add $2 e^{x} \sin (x)$ and $4 e^{x} \sin (x)$ .

$f'' (x) = 6 e^{x} \sin (x) + 2 e^{x} \cos (x)$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $6 e^{x} \sin (x) + 2 e^{x} \cos (x)$ .

$6 e^{x} \sin (x) + 2 e^{x} \cos (x)$