Calculus Examples

$y = x + \sin (x)$ , $(π, π)$

Step 1

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

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Step 1.1

Differentiate.

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Step 1.1.1

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

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

Step 1.1.2

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

$1 + \frac{d}{d x} [\sin (x)]$

Step 1.2

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

$1 + \cos (x)$

Step 1.3

Evaluate the derivative at $x = π$ .

$1 + \cos (π)$

Step 1.4

Simplify.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

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.

$1 - \cos (0)$

Step 1.4.1.2

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

$1 - 1 \cdot 1$

Step 1.4.1.3

Multiply $- 1$ by $1$ .

$1 - 1$

Step 1.4.2

Subtract $1$ from $1$ .

$0$

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 $0$ and a given point $(π, π)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (π) = 0 \cdot (x - (π))$

Step 2.2

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

$y - π = 0 \cdot (x - π)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $0$ by $x - π$ .

$y - π = 0$

Step 2.3.2

Add $π$ to both sides of the equation.

$y = π$

Step 3