Calculus Examples

$\sin^{2} (π x)$

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 $g (x) = \sin (π x)$ .

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Step 1.1

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

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

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 $n = 2$ .

$2 u_{1} \frac{d}{d x} [\sin (π x)]$

Step 1.3

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

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

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 $g (x) = π x$ .

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Step 2.1

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

$2 \sin (π x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [π x])$

Step 2.2

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

$2 \sin (π x) (\cos (u_{2}) \frac{d}{d x} [π x])$

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

$2 \sin (π x) \cos (π x) (π \cdot 1)$

Step 3.3

Multiply $π$ by $1$ .

$2 \sin (π x) \cos (π x) π$

Step 4

Simplify.

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Step 4.1

Reorder the factors of $2 \sin (π x) \cos (π x) π$ .

$2 π \cos (π x) \sin (π x)$

Step 4.2

Add parentheses.

$2 π (\cos (π x) \sin (π x))$

Step 4.3

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

$\cos (π x) \sin (π x) (2 π)$

Step 4.4

Add parentheses.

$\cos (π x) (\sin (π x) \cdot 2) π$

Step 4.5

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

$\sin (π x) \cdot 2 \cos (π x) π$

Step 4.6

Reorder $\sin (π x)$ and $2$ .

$2 \cdot \sin (π x) \cos (π x) π$

Step 4.7

Apply the sine double-angle identity.

$\sin (2 (π x)) π$

Step 4.8

Reorder factors in $\sin (2 π x) π$ .

$π \sin (2 π x)$