Calculus Examples

$y = e^{6 x \sin (2 x)}$

Step 1

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 $g (x) = 6 x \sin (2 x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $6 x \sin (2 x)$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [6 x \sin (2 x)]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [6 x \sin (2 x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $6 x \sin (2 x)$ .

$e^{6 x \sin (2 x)} \frac{d}{d x} [6 x \sin (2 x)]$

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x \sin (2 x)$ with respect to $x$ is $6 \frac{d}{d x} [x \sin (2 x)]$ .

$e^{6 x \sin (2 x)} (6 \frac{d}{d x} [x \sin (2 x)])$

Step 2.2

Move $6$ to the left of $e^{6 x \sin (2 x)}$ .

$6 \cdot e^{6 x \sin (2 x)} \frac{d}{d x} [x \sin (2 x)]$

Step 3

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 $f (x) = x$ and $g (x) = \sin (2 x)$ .

$6 e^{6 x \sin (2 x)} (x \frac{d}{d x} [\sin (2 x)] + \sin (2 x) \frac{d}{d x} [x])$

Step 4

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 $g (x) = 2 x$ .

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Step 4.1

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

$6 e^{6 x \sin (2 x)} (x (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [x])$

Step 4.2

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

$6 e^{6 x \sin (2 x)} (x (\cos (u_{2}) \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [x])$

Step 4.3

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

$6 e^{6 x \sin (2 x)} (x (\cos (2 x) \frac{d}{d x} [2 x]) + \sin (2 x) \frac{d}{d x} [x])$

Step 5

Differentiate.

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Step 5.1

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

$6 e^{6 x \sin (2 x)} (x (\cos (2 x) (2 \frac{d}{d x} [x])) + \sin (2 x) \frac{d}{d x} [x])$

Step 5.2

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

$6 e^{6 x \sin (2 x)} (x (\cos (2 x) (2 \cdot 1)) + \sin (2 x) \frac{d}{d x} [x])$

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $2$ by $1$ .

$6 e^{6 x \sin (2 x)} (x (\cos (2 x) \cdot 2) + \sin (2 x) \frac{d}{d x} [x])$

Step 5.3.2

Move $2$ to the left of $\cos (2 x)$ .

$6 e^{6 x \sin (2 x)} (x (2 \cdot \cos (2 x)) + \sin (2 x) \frac{d}{d x} [x])$

Step 5.4

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

$6 e^{6 x \sin (2 x)} (x (2 \cos (2 x)) + \sin (2 x) \cdot 1)$

Step 5.5

Multiply $\sin (2 x)$ by $1$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $2$ by $6$ .

$12 e^{6 x \sin (2 x)} x \cos (2 x) + 6 e^{6 x \sin (2 x)} \sin (2 x)$