Calculus Examples

$\sin^{2} (2 x^{3} + 3 x)$

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{3}$ with respect to $x$ is $2 \frac{d}{d x} [x^{3}]$ .

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

Step 3.3

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

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

Step 3.4

Multiply $3$ by $2$ .

$2 \sin (2 x^{3} + 3 x) \cos (2 x^{3} + 3 x) (6 x^{2} + \frac{d}{d x} [3 x])$

Step 3.5

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

$2 \sin (2 x^{3} + 3 x) \cos (2 x^{3} + 3 x) (6 x^{2} + 3 \frac{d}{d x} [x])$

Step 3.6

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

$2 \sin (2 x^{3} + 3 x) \cos (2 x^{3} + 3 x) (6 x^{2} + 3 \cdot 1)$

Step 3.7

Multiply $3$ by $1$ .

$2 \sin (2 x^{3} + 3 x) \cos (2 x^{3} + 3 x) (6 x^{2} + 3)$

Step 4

Simplify.

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Step 4.1

Reorder the factors of $2 \sin (2 x^{3} + 3 x) \cos (2 x^{3} + 3 x) (6 x^{2} + 3)$ .

$2 \cos (2 x^{3} + 3 x) \sin (2 x^{3} + 3 x) (6 x^{2} + 3)$

Step 4.2

Reorder $2 \cos (2 x^{3} + 3 x)$ and $\sin (2 x^{3} + 3 x)$ .

$\sin (2 x^{3} + 3 x) (2 \cos (2 x^{3} + 3 x)) (6 x^{2} + 3)$

Step 4.3

Reorder $\sin (2 x^{3} + 3 x)$ and $2$ .

$2 \cdot \sin (2 x^{3} + 3 x) \cos (2 x^{3} + 3 x) (6 x^{2} + 3)$

Step 4.4

Apply the sine double-angle identity.

$\sin (2 (2 x^{3} + 3 x)) (6 x^{2} + 3)$

Step 4.5

Apply the distributive property.

$\sin (2 (2 x^{3}) + 2 (3 x)) (6 x^{2} + 3)$

Step 4.6

Multiply $2$ by $2$ .

$\sin (4 x^{3} + 2 (3 x)) (6 x^{2} + 3)$

Step 4.7

Multiply $3$ by $2$ .

$\sin (4 x^{3} + 6 x) (6 x^{2} + 3)$

Step 4.8

Apply the distributive property.

$\sin (4 x^{3} + 6 x) (6 x^{2}) + \sin (4 x^{3} + 6 x) \cdot 3$

Step 4.9

Rewrite using the commutative property of multiplication.

$6 \sin (4 x^{3} + 6 x) x^{2} + \sin (4 x^{3} + 6 x) \cdot 3$

Step 4.10

Move $3$ to the left of $\sin (4 x^{3} + 6 x)$ .

$6 \sin (4 x^{3} + 6 x) x^{2} + 3 \cdot \sin (4 x^{3} + 6 x)$

Step 4.11

Reorder factors in $6 \sin (4 x^{3} + 6 x) x^{2} + 3 \sin (4 x^{3} + 6 x)$ .

$6 x^{2} \sin (4 x^{3} + 6 x) + 3 \sin (4 x^{3} + 6 x)$