Calculus Examples

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

Step 1

Find the derivative.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

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

$x \cos (x) + \sin (x) \cdot 1$

Step 1.3.2

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

$x \cos (x) + \sin (x)$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0, - 2.02875783, 2.02875783, - 4.91318043, 4.91318043, - 7.97866571, 7.97866571$

Step 3

Solve the original function $f (x) = x \sin (x)$ at $x = 0$ .

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Step 3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = (0) \sin (0)$

Step 3.2

Simplify the result.

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Step 3.2.1

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

$f (0) = 0 \cdot 0$

Step 3.2.2

Multiply $0$ by $0$ .

$f (0) = 0$

Step 3.2.3

The final answer is $0$ .

$0$

Step 4

Solve the original function $f (x) = x \sin (x)$ at $x = - 2.02875783$ .

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Step 4.1

Replace the variable $x$ with $- 2.02875783$ in the expression.

$f (- 2.02875783) = (- 2.02875783) \sin (- 2.02875783)$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $- 2.02875783$ by $\sin (- 2.02875783)$ .

$f (- 2.02875783) = 1.81970574$

Step 4.2.2

The final answer is $1.81970574$ .

$1.81970574$

Step 5

Solve the original function $f (x) = x \sin (x)$ at $x = 2.02875783$ .

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Step 5.1

Replace the variable $x$ with $2.02875783$ in the expression.

$f (2.02875783) = (2.02875783) \sin (2.02875783)$

Step 5.2

Simplify the result.

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Step 5.2.1

Multiply $2.02875783$ by $\sin (2.02875783)$ .

$f (2.02875783) = 1.81970574$

Step 5.2.2

The final answer is $1.81970574$ .

$1.81970574$

Step 6

Solve the original function $f (x) = x \sin (x)$ at $x = - 4.91318043$ .

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Step 6.1

Replace the variable $x$ with $- 4.91318043$ in the expression.

$f (- 4.91318043) = (- 4.91318043) \sin (- 4.91318043)$

Step 6.2

Simplify the result.

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Step 6.2.1

Multiply $- 4.91318043$ by $\sin (- 4.91318043)$ .

$f (- 4.91318043) = - 4.81446988$

Step 6.2.2

The final answer is $- 4.81446988$ .

$- 4.81446988$

Step 7

Solve the original function $f (x) = x \sin (x)$ at $x = 4.91318043$ .

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Step 7.1

Replace the variable $x$ with $4.91318043$ in the expression.

$f (4.91318043) = (4.91318043) \sin (4.91318043)$

Step 7.2

Simplify the result.

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Step 7.2.1

Multiply $4.91318043$ by $\sin (4.91318043)$ .

$f (4.91318043) = - 4.81446988$

Step 7.2.2

The final answer is $- 4.81446988$ .

$- 4.81446988$

Step 8

Solve the original function $f (x) = x \sin (x)$ at $x = - 7.97866571$ .

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Step 8.1

Replace the variable $x$ with $- 7.97866571$ in the expression.

$f (- 7.97866571) = (- 7.97866571) \sin (- 7.97866571)$

Step 8.2

Simplify the result.

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Step 8.2.1

Multiply $- 7.97866571$ by $\sin (- 7.97866571)$ .

$f (- 7.97866571) = 7.91672737$

Step 8.2.2

The final answer is $7.91672737$ .

$7.91672737$

Step 9

Solve the original function $f (x) = x \sin (x)$ at $x = 7.97866571$ .

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Step 9.1

Replace the variable $x$ with $7.97866571$ in the expression.

$f (7.97866571) = (7.97866571) \sin (7.97866571)$

Step 9.2

Simplify the result.

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Step 9.2.1

Multiply $7.97866571$ by $\sin (7.97866571)$ .

$f (7.97866571) = 7.91672737$

Step 9.2.2

The final answer is $7.91672737$ .

$7.91672737$

Step 10

The horizontal tangent lines on function $f (x) = x \sin (x)$ are $y = 0, y = 1.81970574, y = 1.81970574, y = - 4.81446988, y = - 4.81446988, y = 7.91672737, y = 7.91672737$ .

$y = 0, y = 1.81970574, y = 1.81970574, y = - 4.81446988, y = - 4.81446988, y = 7.91672737, y = 7.91672737$

Step 11