Calculus Examples

$\frac{\sin (5 x)}{x}$

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 $f (x) = \sin (5 x)$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [\sin (5 x)] - \sin (5 x) \frac{d}{d x} [x]}{x^{2}}$

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 $g (x) = 5 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $5 x$ .

$\frac{x (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [5 x]) - \sin (5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2

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

$\frac{x (\cos (u) \frac{d}{d x} [5 x]) - \sin (5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 2.3

Replace all occurrences of $u$ with $5 x$ .

$\frac{x (\cos (5 x) \frac{d}{d x} [5 x]) - \sin (5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3

Differentiate.

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{x (\cos (5 x) (5 \frac{d}{d x} [x])) - \sin (5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

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

$\frac{x (\cos (5 x) (5 \cdot 1)) - \sin (5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $5$ by $1$ .

$\frac{x (\cos (5 x) \cdot 5) - \sin (5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3.2

Move $5$ to the left of $\cos (5 x)$ .

$\frac{x (5 \cdot \cos (5 x)) - \sin (5 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.4

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

$\frac{x (5 \cos (5 x)) - \sin (5 x) \cdot 1}{x^{2}}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Multiply $- 1$ by $1$ .

$\frac{x (5 \cos (5 x)) - \sin (5 x)}{x^{2}}$

Step 3.5.2

Rewrite using the commutative property of multiplication.

$\frac{5 x \cos (5 x) - \sin (5 x)}{x^{2}}$