Calculus Examples

$y = 2 x \cos (x)$ , $(π, - 2 π)$

Step 1

Find the first derivative and evaluate at $x = π$ and $y = - 2 π$ to find the slope of the tangent line.

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Step 1.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x \cos (x)$ with respect to $x$ is $2 \frac{d}{d x} [x \cos (x)]$ .

$2 \frac{d}{d x} [x \cos (x)]$

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

$2 (x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x])$

Step 1.3

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

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

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 $n = 1$ .

$2 (x (- \sin (x)) + \cos (x) \cdot 1)$

Step 1.4.2

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

$2 (x (- \sin (x)) + \cos (x))$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$2 (x (- \sin (x))) + 2 \cos (x)$

Step 1.5.2

Multiply $- 1$ by $2$ .

$- 2 x \sin (x) + 2 \cos (x)$

Step 1.6

Evaluate the derivative at $x = π$ .

$- 2 π \sin (π) + 2 \cos (π)$

Step 1.7

Simplify.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$- 2 π \sin (0) + 2 \cos (π)$

Step 1.7.1.2

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

$- 2 π \cdot 0 + 2 \cos (π)$

Step 1.7.1.3

Multiply $- 2 π \cdot 0$ .

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Step 1.7.1.3.1

Multiply $0$ by $- 2$ .

$0 π + 2 \cos (π)$

Step 1.7.1.3.2

Multiply $0$ by $π$ .

$0 + 2 \cos (π)$

Step 1.7.1.4

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$0 + 2 (- \cos (0))$

Step 1.7.1.5

The exact value of $\cos (0)$ is $1$ .

$0 + 2 (- 1 \cdot 1)$

Step 1.7.1.6

Multiply $2 (- 1 \cdot 1)$ .

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Step 1.7.1.6.1

Multiply $- 1$ by $1$ .

$0 + 2 \cdot - 1$

Step 1.7.1.6.2

Multiply $2$ by $- 1$ .

$0 - 2$

Step 1.7.2

Subtract $2$ from $0$ .

$- 2$

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 $- 2$ and a given point $(π, - 2 π)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 2 π) = - 2 \cdot (x - (π))$

Step 2.2

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

$y + 2 π = - 2 \cdot (x - π)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 2 \cdot (x - π)$ .

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Step 2.3.1.1

Rewrite.

$y + 2 π = 0 + 0 - 2 \cdot (x - π)$

Step 2.3.1.2

Simplify by adding zeros.

$y + 2 π = - 2 \cdot (x - π)$

Step 2.3.1.3

Apply the distributive property.

$y + 2 π = - 2 x - 2 (- π)$

Step 2.3.1.4

Multiply $- 1$ by $- 2$ .

$y + 2 π = - 2 x + 2 π$

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

Subtract $2 π$ from both sides of the equation.

$y = - 2 x + 2 π - 2 π$

Step 2.3.2.2

Combine the opposite terms in $- 2 x + 2 π - 2 π$ .

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Step 2.3.2.2.1

Subtract $2 π$ from $2 π$ .

$y = - 2 x + 0$

Step 2.3.2.2.2

Add $- 2 x$ and $0$ .

$y = - 2 x$

Step 3