Calculus Examples

\frac{sin (x^{2})}{3 x}

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

Step 1

Since

\frac{1}{3}

$\frac{1}{3}$ is constant with respect to

x

$x$ , the derivative of

\frac{sin (x^{2})}{3 x}

$\frac{\sin (x^{2})}{3 x}$ with respect to

x

$x$ is

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

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

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

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

Step 2

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

$\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}

$\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where

f (x) = sin (x^{2})

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

g (x) = x

$g (x) = x$ .

\frac{1}{3} \cdot \frac{x \frac{d}{d x} [sin (x^{2})] - sin (x^{2}) \frac{d}{d x} [x]}{x^{2}}

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

Step 3

Differentiate using the chain rule, which states that

\frac{d}{d x} [f (g (x))]

$\frac{d}{d x} [f (g (x))]$ is

$f' (g (x)) g' (x)$ where

$f (x) = \sin (x)$ and

$g (x) = x^{2}$ .

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Step 3.1

To apply the Chain Rule, set

$u$ as

$x^{2}$ .

$\frac{1}{3} \cdot \frac{x (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x^{2}]) - \sin (x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

The derivative of

$\sin (u)$ with respect to

$u$ is

$\cos (u)$ .

$\frac{1}{3} \cdot \frac{x (\cos (u) \frac{d}{d x} [x^{2}]) - \sin (x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

Replace all occurrences of

$u$ with

$x^{2}$ .

$\frac{1}{3} \cdot \frac{x (\cos (x^{2}) \frac{d}{d x} [x^{2}]) - \sin (x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 4

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$\frac{1}{3} \cdot \frac{x (\cos (x^{2}) (2 x)) - \sin (x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 5

Raise

$x$ to the power of

$1$ .

$\frac{1}{3} \cdot \frac{x^{1} x (\cos (x^{2}) \cdot (2)) - \sin (x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 6

Raise

$x$ to the power of

$1$ .

$\frac{1}{3} \cdot \frac{x^{1} x^{1} (\cos (x^{2}) \cdot (2)) - \sin (x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 7

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 8

Simplify the expression.

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Step 8.1

Add

$1$ and

$1$ .

$\frac{1}{3} \cdot \frac{x^{2} (\cos (x^{2}) \cdot (2)) - \sin (x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 8.2

Move

$2$ to the left of

$\cos (x^{2})$ .

$\frac{1}{3} \cdot \frac{x^{2} (2 \cdot \cos (x^{2})) - \sin (x^{2}) \frac{d}{d x} [x]}{x^{2}}$

Step 9

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$\frac{1}{3} \cdot \frac{x^{2} (2 \cos (x^{2})) - \sin (x^{2}) \cdot 1}{x^{2}}$

Step 10

Combine fractions.

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Step 10.1

Multiply

$- 1$ by

$1$ .

$\frac{1}{3} \cdot \frac{x^{2} (2 \cos (x^{2})) - \sin (x^{2})}{x^{2}}$

Step 10.2

Multiply

$\frac{1}{3}$ by

$\frac{x^{2} (2 \cos (x^{2})) - \sin (x^{2})}{x^{2}}$ .

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

Step 10.3

Rewrite using the commutative property of multiplication.

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