Calculus Examples

$f (x) = e^{2 x} \cos^{3} (2 x^{2} - 3)$

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 $f (x) = e^{2 x}$ and $g (x) = \cos^{3} (2 x^{2} - 3)$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = \cos (2 x^{2} - 3)$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\cos (2 x^{2} - 3)$ .

$e^{2 x} (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [\cos (2 x^{2} - 3)]) + \cos^{3} (2 x^{2} - 3) \frac{d}{d x} [e^{2 x}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$e^{2 x} (3 {u_{1}}^{2} \frac{d}{d x} [\cos (2 x^{2} - 3)]) + \cos^{3} (2 x^{2} - 3) \frac{d}{d x} [e^{2 x}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $\cos (2 x^{2} - 3)$ .

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

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 $g (x) = 2 x^{2} - 3$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $2 x^{2} - 3$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u_{2}$ with $2 x^{2} - 3$ .

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

Step 4

Differentiate.

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Step 4.1

Multiply $- 1$ by $3$ .

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

Step 4.2

By the Sum Rule, the derivative of $2 x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 3]$ .

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

Step 4.3

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

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

Step 4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 4.5

Multiply $2$ by $2$ .

$e^{2 x} (- 3 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) (4 x + \frac{d}{d x} [- 3])) + \cos^{3} (2 x^{2} - 3) \frac{d}{d x} [e^{2 x}]$

Step 4.6

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$e^{2 x} (- 3 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) (4 x + 0)) + \cos^{3} (2 x^{2} - 3) \frac{d}{d x} [e^{2 x}]$

Step 4.7

Simplify the expression.

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Step 4.7.1

Add $4 x$ and $0$ .

$e^{2 x} (- 3 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) (4 x)) + \cos^{3} (2 x^{2} - 3) \frac{d}{d x} [e^{2 x}]$

Step 4.7.2

Multiply $4$ by $- 3$ .

$e^{2 x} (- 12 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) x) + \cos^{3} (2 x^{2} - 3) \frac{d}{d x} [e^{2 x}]$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 5.1

To apply the Chain Rule, set $u_{3}$ as $2 x$ .

$e^{2 x} (- 12 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) x) + \cos^{3} (2 x^{2} - 3) (\frac{d}{d u_{3}} [e^{u_{3}}] \frac{d}{d x} [2 x])$

Step 5.2

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

$e^{2 x} (- 12 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) x) + \cos^{3} (2 x^{2} - 3) (e^{u_{3}} \frac{d}{d x} [2 x])$

Step 5.3

Replace all occurrences of $u_{3}$ with $2 x$ .

$e^{2 x} (- 12 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) x) + \cos^{3} (2 x^{2} - 3) (e^{2 x} \frac{d}{d x} [2 x])$

Step 6

Differentiate.

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Step 6.1

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

$e^{2 x} (- 12 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) x) + \cos^{3} (2 x^{2} - 3) (e^{2 x} (2 \frac{d}{d x} [x]))$

Step 6.2

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

$e^{2 x} (- 12 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) x) + \cos^{3} (2 x^{2} - 3) (e^{2 x} (2 \cdot 1))$

Step 6.3

Simplify the expression.

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Step 6.3.1

Multiply $2$ by $1$ .

$e^{2 x} (- 12 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) x) + \cos^{3} (2 x^{2} - 3) (e^{2 x} \cdot 2)$

Step 6.3.2

Move $2$ to the left of $e^{2 x}$ .

$e^{2 x} (- 12 \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) x) + \cos^{3} (2 x^{2} - 3) (2 \cdot e^{2 x})$

Step 7

Simplify.

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Step 7.1

Reorder terms.

$- 12 e^{2 x} x \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) + 2 e^{2 x} \cos^{3} (2 x^{2} - 3)$

Step 7.2

Reorder factors in $- 12 e^{2 x} x \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) + 2 e^{2 x} \cos^{3} (2 x^{2} - 3)$ .

$- 12 x e^{2 x} \cos^{2} (2 x^{2} - 3) \sin (2 x^{2} - 3) + 2 e^{2 x} \cos^{3} (2 x^{2} - 3)$