Calculus Examples

$y = 8 x \sin (x)$ , $(\frac{π}{2}, 4 π)$

Step 1

Find the first derivative and evaluate at $x = \frac{π}{2}$ and $y = 4 π$ to find the slope of the tangent line.

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Step 1.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x \sin (x)$ with respect to $x$ is $8 \frac{d}{d x} [x \sin (x)]$ .

$8 \frac{d}{d x} [x \sin (x)]$

Step 1.2

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

$8 (x \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x])$

Step 1.3

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

$8 (x \cos (x) + \sin (x) \frac{d}{d x} [x])$

Step 1.4

Differentiate using the Power Rule.

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Step 1.4.1

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

$8 (x \cos (x) + \sin (x) \cdot 1)$

Step 1.4.2

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

$8 (x \cos (x) + \sin (x))$

Step 1.5

Apply the distributive property.

$8 x \cos (x) + 8 \sin (x)$

Step 1.6

Evaluate the derivative at $x = \frac{π}{2}$ .

$8 (\frac{π}{2}) \cos (\frac{π}{2}) + 8 \sin (\frac{π}{2})$

Step 1.7

Simplify.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Cancel the common factor of $2$ .

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Step 1.7.1.1.1

Factor $2$ out of $8$ .

$2 (4) \frac{π}{2} \cos (\frac{π}{2}) + 8 \sin (\frac{π}{2})$

Step 1.7.1.1.2

Cancel the common factor.

$2 \cdot 4 \frac{π}{2} \cos (\frac{π}{2}) + 8 \sin (\frac{π}{2})$

Step 1.7.1.1.3

Rewrite the expression.

$4 π \cos (\frac{π}{2}) + 8 \sin (\frac{π}{2})$

Step 1.7.1.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$4 π \cdot 0 + 8 \sin (\frac{π}{2})$

Step 1.7.1.3

Multiply $4 π \cdot 0$ .

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Step 1.7.1.3.1

Multiply $0$ by $4$ .

$0 π + 8 \sin (\frac{π}{2})$

Step 1.7.1.3.2

Multiply $0$ by $π$ .

$0 + 8 \sin (\frac{π}{2})$

Step 1.7.1.4

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$0 + 8 \cdot 1$

Step 1.7.1.5

Multiply $8$ by $1$ .

$0 + 8$

Step 1.7.2

Add $0$ and $8$ .

$8$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $8$ and a given point $(\frac{π}{2}, 4 π)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (4 π) = 8 \cdot (x - (\frac{π}{2}))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - 4 π = 8 \cdot (x - \frac{π}{2})$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $8 \cdot (x - \frac{π}{2})$ .

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Step 2.3.1.1

Rewrite.

$y - 4 π = 0 + 0 + 8 \cdot (x - \frac{π}{2})$

Step 2.3.1.2

Simplify by adding zeros.

$y - 4 π = 8 \cdot (x - \frac{π}{2})$

Step 2.3.1.3

Apply the distributive property.

$y - 4 π = 8 x + 8 (- \frac{π}{2})$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$y - 4 π = 8 x + 8 \frac{- π}{2}$

Step 2.3.1.4.2

Factor $2$ out of $8$ .

$y - 4 π = 8 x + 2 (4) \frac{- π}{2}$

Step 2.3.1.4.3

Cancel the common factor.

$y - 4 π = 8 x + 2 \cdot 4 \frac{- π}{2}$

Step 2.3.1.4.4

Rewrite the expression.

$y - 4 π = 8 x + 4 (- π)$

Step 2.3.1.5

Multiply $- 1$ by $4$ .

$y - 4 π = 8 x - 4 π$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.3.2.1

Add $4 π$ to both sides of the equation.

$y = 8 x - 4 π + 4 π$

Step 2.3.2.2

Combine the opposite terms in $8 x - 4 π + 4 π$ .

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Step 2.3.2.2.1

Add $- 4 π$ and $4 π$ .

$y = 8 x + 0$

Step 2.3.2.2.2

Add $8 x$ and $0$ .

$y = 8 x$

Step 3