Calculus Examples

$y = \sin^{2} (x) (x^{3})$

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 $f (x) = \sin^{2} (x)$ and $g (x) = x^{3}$ .

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

Step 2

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

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

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u$ as $\sin (x)$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Reorder terms.

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

Step 5.2

Simplify each term.

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Step 5.2.1

Add parentheses.

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

Step 5.2.2

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

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

Step 5.2.3

Add parentheses.

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

Step 5.2.4

Reorder $\cos (x)$ and $\sin (x) \cdot 2$ .

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

Step 5.2.5

Reorder $\sin (x)$ and $2$ .

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

Step 5.2.6

Apply the sine double-angle identity.

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

Step 5.3

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

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