Calculus Examples

$\sin (x) - x \cos (x)$

Step 1

Write $\sin (x) - x \cos (x)$ as a function.

$f (x) = \sin (x) - x \cos (x)$

Step 2

Find the first derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $\sin (x) - x \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- x \cos (x)]$ .

$f' (x) = \frac{d}{d x} (\sin (x)) + \frac{d}{d x} (- x \cos (x))$

Step 2.1.2

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

$f' (x) = \cos (x) + \frac{d}{d x} (- x \cos (x))$

Step 2.1.3

Evaluate $\frac{d}{d x} [- x \cos (x)]$ .

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Step 2.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x \cos (x)$ with respect to $x$ is $- \frac{d}{d x} [x \cos (x)]$ .

$f' (x) = \cos (x) - \frac{d}{d x} (x \cos (x))$

Step 2.1.3.2

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 $g (x) = \cos (x)$ .

$f' (x) = \cos (x) - (x \frac{d}{d x} (\cos (x)) + \cos (x) \frac{d}{d x} (x))$

Step 2.1.3.3

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

$f' (x) = \cos (x) - (x (- \sin (x)) + \cos (x) \frac{d}{d x} (x))$

Step 2.1.3.4

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

$f' (x) = \cos (x) - (x (- \sin (x)) + \cos (x) \cdot 1)$

Step 2.1.3.5

Multiply $\cos (x)$ by $1$ .

$f' (x) = \cos (x) - (x (- \sin (x)) + \cos (x))$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Apply the distributive property.

$f' (x) = \cos (x) - (x (- \sin (x))) - \cos (x)$

Step 2.1.4.2

Combine terms.

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Step 2.1.4.2.1

Multiply $- 1$ by $- 1$ .

$f' (x) = \cos (x) + 1 (x (\sin (x))) - \cos (x)$

Step 2.1.4.2.2

Multiply $x$ by $1$ .

$f' (x) = \cos (x) + x \sin (x) - \cos (x)$

Step 2.1.4.2.3

Subtract $\cos (x)$ from $\cos (x)$ .

$f' (x) = x \sin (x) + 0$

Step 2.1.4.2.4

Add $x \sin (x)$ and $0$ .

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

Step 2.2

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

$x \sin (x)$

Step 3

Set the first derivative equal to $0$ then solve the equation $x \sin (x) = 0$ .

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Step 3.1

Set the first derivative equal to $0$ .

$x \sin (x) = 0$

Step 3.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$\sin (x) = 0$

Step 3.3

Set $x$ equal to $0$ .

$x = 0$

Step 3.4

Set $\sin (x)$ equal to $0$ and solve for $x$ .

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Step 3.4.1

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Step 3.4.2

Solve $\sin (x) = 0$ for $x$ .

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Step 3.4.2.1

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$x = \arcsin (0)$

Step 3.4.2.2

Simplify the right side.

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Step 3.4.2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

Step 3.4.2.3

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - 0$

Step 3.4.2.4

Subtract $0$ from $π$ .

$x = π$

Step 3.4.2.5

Find the period of $\sin (x)$ .

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Step 3.4.2.5.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.4.2.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.4.2.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.4.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.4.2.6

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 3.5

The final solution is all the values that make $x \sin (x) = 0$ true.

$x = 0, 2 π n, π + 2 π n$ , for any integer $n$

Step 3.6

Consolidate the answers.

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Step 3.6.1

Consolidate $0$ and $2 π n$ to $2 π n$ .

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 3.6.2

Consolidate the answers.

$x = π n$ , for any integer $n$

Step 4

The values which make the derivative equal to $0$ are $π n$ .

$π n$

Step 5

After finding the point that makes the derivative $f' (x) = x \sin (x)$ equal to $0$ or undefined, the interval to check where $f (x) = \sin (x) - x \cos (x)$ is increasing and where it is decreasing is $(- \infty, π n) \cup (π n, \infty)$ .

$(- \infty, π n) \cup (π n, \infty)$

Step 6

Substitute a value from the interval $(- \infty, π n)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $π n - 1$ in the expression.

$f' (π n - 1) = (π n - 1) \sin (π n - 1)$

Step 6.2

Simplify the result.

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Step 6.2.1

Apply the distributive property.

$f' (π n - 1) = π n \sin (π n - 1) - 1 \sin (π n - 1)$

Step 6.2.2

Rewrite $- 1 \sin (π n - 1)$ as $- \sin (π n - 1)$ .

$f' (π n - 1) = π n \sin (π n - 1) - \sin (π n - 1)$

Step 6.2.3

The final answer is $π n \sin (π n - 1) - \sin (π n - 1)$ .

$π n \sin (π n - 1) - \sin (π n - 1)$

Step 6.3

At $x = π n - 1$ the derivative is $π n \sin (π n - 1) - \sin (π n - 1)$ . Since this is negative, the function is decreasing on $(- \infty, π n)$ .

Decreasing on $(- \infty, π n)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(π n, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $π n + 1$ in the expression.

$f' (π n + 1) = (π n + 1) \sin (π n + 1)$

Step 7.2

Simplify the result.

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Step 7.2.1

Apply the distributive property.

$f' (π n + 1) = π n \sin (π n + 1) + 1 \sin (π n + 1)$

Step 7.2.2

Multiply $\sin (π n + 1)$ by $1$ .

$f' (π n + 1) = π n \sin (π n + 1) + \sin (π n + 1)$

Step 7.2.3

The final answer is $π n \sin (π n + 1) + \sin (π n + 1)$ .

$π n \sin (π n + 1) + \sin (π n + 1)$

Step 7.3

At $x = π n + 1$ the derivative is $π n \sin (π n + 1) + \sin (π n + 1)$ . Since this is positive, the function is increasing on $(π n, \infty)$ .

Increasing on $(π n, \infty)$ since $f' (x) > 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(π n, \infty)$

Decreasing on: $(- \infty, π n)$

Step 9