Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sin (3 x)$ and $g (x) = x^{2}$ .

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

Step 2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [\sin (3 x)] - \sin (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u$ as $3 x$ .

$\frac{x^{2} (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [3 x]) - \sin (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.2

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

$\frac{x^{2} (\cos (u) \frac{d}{d x} [3 x]) - \sin (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.3

Replace all occurrences of $u$ with $3 x$ .

$\frac{x^{2} (\cos (3 x) \frac{d}{d x} [3 x]) - \sin (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4

Differentiate.

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Step 4.1

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

$\frac{x^{2} (\cos (3 x) (3 \frac{d}{d x} [x])) - \sin (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.2

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

$\frac{x^{2} (\cos (3 x) (3 \cdot 1)) - \sin (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $3$ by $1$ .

$\frac{x^{2} (\cos (3 x) \cdot 3) - \sin (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.3.2

Move $3$ to the left of $\cos (3 x)$ .

$\frac{x^{2} (3 \cdot \cos (3 x)) - \sin (3 x) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 4.4

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

$\frac{x^{2} (3 \cos (3 x)) - \sin (3 x) (2 x)}{x^{4}}$

Step 4.5

Simplify with factoring out.

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Step 4.5.1

Multiply $2$ by $- 1$ .

$\frac{x^{2} (3 \cos (3 x)) - 2 \sin (3 x) x}{x^{4}}$

Step 4.5.2

Factor $x$ out of $x^{2} (3 \cos (3 x)) - 2 \sin (3 x) x$ .

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Step 4.5.2.1

Factor $x$ out of $x^{2} (3 \cos (3 x))$ .

$\frac{x (x (3 \cos (3 x))) - 2 \sin (3 x) x}{x^{4}}$

Step 4.5.2.2

Factor $x$ out of $- 2 \sin (3 x) x$ .

$\frac{x (x (3 \cos (3 x))) + x (- 2 \sin (3 x))}{x^{4}}$

Step 4.5.2.3

Factor $x$ out of $x (x (3 \cos (3 x))) + x (- 2 \sin (3 x))$ .

$\frac{x (x (3 \cos (3 x)) - 2 \sin (3 x))}{x^{4}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor $x$ out of $x^{4}$ .

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

Step 5.2

Cancel the common factor.

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

Step 5.3

Rewrite the expression.

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

Step 6

Rewrite using the commutative property of multiplication.

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