Calculus Examples

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

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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

Step 4

Differentiate using the Power Rule.

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Step 4.1

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

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

Step 4.2

Multiply $\sin^{2} (x)$ by $1$ .

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

Step 5

Simplify.

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Step 5.1

Reorder terms.

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

Step 5.2

Simplify each term.

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Step 5.2.1

Add parentheses.

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

Step 5.2.2

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

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

Step 5.2.3

Add parentheses.

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

Step 5.2.4

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

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

Step 5.2.5

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

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

Step 5.2.6

Apply the sine double-angle identity.

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

Step 5.3

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

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