Calculus Examples

y = \frac{1 + sin (x)}{cos (x)}

$y = \frac{1 + \sin (x)}{\cos (x)}$ ,

(π, - 1)

$(π, - 1)$

Step 1

Find the first derivative and evaluate at

x = π

$x = π$ and

y = - 1

$y = - 1$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

$\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}

$\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where

f (x) = 1 + sin (x)

$f (x) = 1 + \sin (x)$ and

g (x) = cos (x)

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

\frac{cos (x) \frac{d}{d x} [1 + sin (x)] - (1 + sin (x)) \frac{d}{d x} [cos (x)]}{{cos}^{2} (x)}

$\frac{\cos (x) \frac{d}{d x} [1 + \sin (x)] - (1 + \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of

1 + sin (x)

$1 + \sin (x)$ with respect to

x

$x$ is

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

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

\frac{cos (x) (\frac{d}{d x} [1] + \frac{d}{d x} [sin (x)]) - (1 + sin (x)) \frac{d}{d x} [cos (x)]}{{cos}^{2} (x)}

$\frac{\cos (x) (\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)]) - (1 + \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 1.2.2

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

\frac{cos (x) (0 + \frac{d}{d x} [sin (x)]) - (1 + sin (x)) \frac{d}{d x} [cos (x)]}{{cos}^{2} (x)}

$\frac{\cos (x) (0 + \frac{d}{d x} [\sin (x)]) - (1 + \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 1.2.3

Add

0

$0$ and

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

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

\frac{cos (x) \frac{d}{d x} [sin (x)] - (1 + sin (x)) \frac{d}{d x} [cos (x)]}{{cos}^{2} (x)}

$\frac{\cos (x) \frac{d}{d x} [\sin (x)] - (1 + \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

\frac{cos (x) \frac{d}{d x} [sin (x)] - (1 + sin (x)) \frac{d}{d x} [cos (x)]}{{cos}^{2} (x)}

$\frac{\cos (x) \frac{d}{d x} [\sin (x)] - (1 + \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 1.3

The derivative of

sin (x)

$\sin (x)$ with respect to

x

$x$ is

cos (x)

$\cos (x)$ .

\frac{cos (x) cos (x) - (1 + sin (x)) \frac{d}{d x} [cos (x)]}{{cos}^{2} (x)}

$\frac{\cos (x) \cos (x) - (1 + \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 1.4

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

\frac{{cos}^{1} (x) cos (x) - (1 + sin (x)) \frac{d}{d x} [cos (x)]}{{cos}^{2} (x)}

$\frac{\cos^{1} (x) \cos (x) - (1 + \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 1.5

Raise

cos (x)

$\cos (x)$ to the power of

1

$1$ .

\frac{{cos}^{1} (x) {cos}^{1} (x) - (1 + sin (x)) \frac{d}{d x} [cos (x)]}{{cos}^{2} (x)}

$\frac{\cos^{1} (x) \cos^{1} (x) - (1 + \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 1.6

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{{cos (x)}^{1 + 1} - (1 + sin (x)) \frac{d}{d x} [cos (x)]}{{cos}^{2} (x)}

$\frac{{\cos (x)}^{1 + 1} - (1 + \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 1.7

Add

1

$1$ and

1

$1$ .

\frac{{cos}^{2} (x) - (1 + sin (x)) \frac{d}{d x} [cos (x)]}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) - (1 + \sin (x)) \frac{d}{d x} [\cos (x)]}{\cos^{2} (x)}$

Step 1.8

The derivative of

cos (x)

$\cos (x)$ with respect to

x

$x$ is

- sin (x)

$- \sin (x)$ .

\frac{{cos}^{2} (x) - (1 + sin (x)) (- sin (x))}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) - (1 + \sin (x)) (- \sin (x))}{\cos^{2} (x)}$

Step 1.9

Multiply.

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Step 1.9.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{{cos}^{2} (x) + 1 (1 + sin (x)) sin (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + 1 (1 + \sin (x)) \sin (x)}{\cos^{2} (x)}$

Step 1.9.2

Multiply

1 + sin (x)

$1 + \sin (x)$ by

1

$1$ .

\frac{{cos}^{2} (x) + (1 + sin (x)) sin (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + (1 + \sin (x)) \sin (x)}{\cos^{2} (x)}$

\frac{{cos}^{2} (x) + (1 + sin (x)) sin (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + (1 + \sin (x)) \sin (x)}{\cos^{2} (x)}$

Step 1.10

Simplify.

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Step 1.10.1

Apply the distributive property.

\frac{{cos}^{2} (x) + 1 sin (x) + sin (x) sin (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + 1 \sin (x) + \sin (x) \sin (x)}{\cos^{2} (x)}$

Step 1.10.2

Simplify the numerator.

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Step 1.10.2.1

Simplify each term.

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Step 1.10.2.1.1

Multiply

sin (x)

$\sin (x)$ by

1

$1$ .

\frac{{cos}^{2} (x) + sin (x) + sin (x) sin (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + \sin (x) + \sin (x) \sin (x)}{\cos^{2} (x)}$

Step 1.10.2.1.2

Multiply

sin (x) sin (x)

$\sin (x) \sin (x)$ .

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Step 1.10.2.1.2.1

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

\frac{{cos}^{2} (x) + sin (x) + {sin}^{1} (x) sin (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + \sin (x) + \sin^{1} (x) \sin (x)}{\cos^{2} (x)}$

Step 1.10.2.1.2.2

Raise

sin (x)

$\sin (x)$ to the power of

1

$1$ .

\frac{{cos}^{2} (x) + sin (x) + {sin}^{1} (x) {sin}^{1} (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + \sin (x) + \sin^{1} (x) \sin^{1} (x)}{\cos^{2} (x)}$

Step 1.10.2.1.2.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{{cos}^{2} (x) + sin (x) + {sin (x)}^{1 + 1}}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + \sin (x) + {\sin (x)}^{1 + 1}}{\cos^{2} (x)}$

Step 1.10.2.1.2.4

Add

1

$1$ and

1

$1$ .

\frac{{cos}^{2} (x) + sin (x) + {sin}^{2} (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + \sin (x) + \sin^{2} (x)}{\cos^{2} (x)}$

\frac{{cos}^{2} (x) + sin (x) + {sin}^{2} (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + \sin (x) + \sin^{2} (x)}{\cos^{2} (x)}$

\frac{{cos}^{2} (x) + sin (x) + {sin}^{2} (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + \sin (x) + \sin^{2} (x)}{\cos^{2} (x)}$

Step 1.10.2.2

Move

{sin}^{2} (x)

$\sin^{2} (x)$ .

\frac{{cos}^{2} (x) + {sin}^{2} (x) + sin (x)}{{cos}^{2} (x)}

$\frac{\cos^{2} (x) + \sin^{2} (x) + \sin (x)}{\cos^{2} (x)}$

Step 1.10.2.3

Rearrange terms.

\frac{{sin}^{2} (x) + {cos}^{2} (x) + sin (x)}{{cos}^{2} (x)}

$\frac{\sin^{2} (x) + \cos^{2} (x) + \sin (x)}{\cos^{2} (x)}$

Step 1.10.2.4

Apply pythagorean identity.

\frac{1 + sin (x)}{{cos}^{2} (x)}

$\frac{1 + \sin (x)}{\cos^{2} (x)}$

\frac{1 + sin (x)}{{cos}^{2} (x)}

$\frac{1 + \sin (x)}{\cos^{2} (x)}$

\frac{1 + sin (x)}{{cos}^{2} (x)}

$\frac{1 + \sin (x)}{\cos^{2} (x)}$

Step 1.11

Evaluate the derivative at

x = π

$x = π$ .

\frac{1 + sin (π)}{{cos}^{2} (π)}

$\frac{1 + \sin (π)}{\cos^{2} (π)}$

Step 1.12

Simplify.

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Step 1.12.1

Simplify the numerator.

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Step 1.12.1.1

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

\frac{1 + sin (0)}{{cos}^{2} (π)}

$\frac{1 + \sin (0)}{\cos^{2} (π)}$

Step 1.12.1.2

The exact value of

sin (0)

$\sin (0)$ is

0

$0$ .

\frac{1 + 0}{{cos}^{2} (π)}

$\frac{1 + 0}{\cos^{2} (π)}$

Step 1.12.1.3

Add

1

$1$ and

0

$0$ .

\frac{1}{{cos}^{2} (π)}

$\frac{1}{\cos^{2} (π)}$

\frac{1}{{cos}^{2} (π)}

$\frac{1}{\cos^{2} (π)}$

Step 1.12.2

Simplify the denominator.

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Step 1.12.2.1

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.

\frac{1}{{(- cos (0))}^{2}}

$\frac{1}{{(- \cos (0))}^{2}}$

Step 1.12.2.2

The exact value of

cos (0)

$\cos (0)$ is

1

$1$ .

\frac{1}{{(- 1 \cdot 1)}^{2}}

$\frac{1}{{(- 1 \cdot 1)}^{2}}$

Step 1.12.2.3

Multiply

- 1

$- 1$ by

1

$1$ .

\frac{1}{{(- 1)}^{2}}

$\frac{1}{{(- 1)}^{2}}$

Step 1.12.2.4

Raise

- 1

$- 1$ to the power of

2

$2$ .

\frac{1}{1}

$\frac{1}{1}$

\frac{1}{1}

$\frac{1}{1}$

Step 1.12.3

Cancel the common factor of

1

$1$ .

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Step 1.12.3.1

Cancel the common factor.

$\frac{1}{1}$

Step 1.12.3.2

Rewrite the expression.

$1$

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

$1$ and a given point

$(π, - 1)$ 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

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = 1 \cdot (x - (π))$

Step 2.2

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

$y + 1 = 1 \cdot (x - π)$

Step 2.3

Solve for

$y$ .

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Step 2.3.1

Multiply

$x - π$ by

$1$ .

$y + 1 = x - π$

Step 2.3.2

Subtract

$1$ from both sides of the equation.

$y = x - π - 1$

Step 3