Calculus Examples

$g (x) = e^{c x} (\cos (a x) - 3 \sin (a x))$

Step 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 $f (x) = e^{c x}$ and $g (x) = \cos (a x) - 3 \sin (a x)$ .

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

Step 2

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

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = a x$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $a x$ .

$e^{c x} (\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [a x] + \frac{d}{d x} [- 3 \sin (a x)]) + (\cos (a x) - 3 \sin (a x)) \frac{d}{d x} [e^{c x}]$

Step 3.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$e^{c x} (- \sin (u_{1}) \frac{d}{d x} [a x] + \frac{d}{d x} [- 3 \sin (a x)]) + (\cos (a x) - 3 \sin (a x)) \frac{d}{d x} [e^{c x}]$

Step 3.3

Replace all occurrences of $u_{1}$ with $a x$ .

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

Step 4

Differentiate.

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Step 4.1

Since $a$ is constant with respect to $x$ , the derivative of $a x$ with respect to $x$ is $a \frac{d}{d x} [x]$ .

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

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{c x} (- \sin (a x) (a \cdot 1) + \frac{d}{d x} [- 3 \sin (a x)]) + (\cos (a x) - 3 \sin (a x)) \frac{d}{d x} [e^{c x}]$

Step 4.3

Multiply $a$ by $1$ .

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

Step 4.4

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

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

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = a x$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $a x$ .

$e^{c x} (- \sin (a x) a - 3 (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [a x])) + (\cos (a x) - 3 \sin (a x)) \frac{d}{d x} [e^{c x}]$

Step 5.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$e^{c x} (- \sin (a x) a - 3 (\cos (u_{2}) \frac{d}{d x} [a x])) + (\cos (a x) - 3 \sin (a x)) \frac{d}{d x} [e^{c x}]$

Step 5.3

Replace all occurrences of $u_{2}$ with $a x$ .

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

Step 6

Differentiate.

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Step 6.1

Since $a$ is constant with respect to $x$ , the derivative of $a x$ with respect to $x$ is $a \frac{d}{d x} [x]$ .

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

Step 6.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{c x} (- \sin (a x) a - 3 \cos (a x) (a \cdot 1)) + (\cos (a x) - 3 \sin (a x)) \frac{d}{d x} [e^{c x}]$

Step 6.3

Multiply $a$ by $1$ .

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

Step 7

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 $g (x) = c x$ .

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Step 7.1

To apply the Chain Rule, set $u_{3}$ as $c x$ .

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

Step 7.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

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

Step 7.3

Replace all occurrences of $u_{3}$ with $c x$ .

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

Step 8

Differentiate.

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Step 8.1

Since $c$ is constant with respect to $x$ , the derivative of $c x$ with respect to $x$ is $c \frac{d}{d x} [x]$ .

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

Step 8.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{c x} (- \sin (a x) a - 3 \cos (a x) a) + (\cos (a x) - 3 \sin (a x)) e^{c x} (c \cdot 1)$

Step 8.3

Multiply $c$ by $1$ .

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

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Apply the distributive property.

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

Step 9.4

Reorder terms.

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