Calculus Examples

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

Step 1

Find the first derivative and evaluate at $x = π$ and $y = - 1$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1 + \sin (x)$ and $g (x) = \cos (x)$ .

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

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $1 + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)]$ .

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

Step 1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.2.3

Add $0$ and $\frac{d}{d x} [\sin (x)]$ .

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

Step 1.3

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

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

Step 1.4

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.5

Raise $\cos (x)$ to the power of $1$ .

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

Step 1.6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.7

Add $1$ and $1$ .

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

Step 1.8

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

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

Step 1.9

Multiply.

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Step 1.9.1

Multiply $- 1$ by $- 1$ .

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

Step 1.9.2

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

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

Step 1.10

Simplify.

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Step 1.10.1

Apply the distributive property.

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

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

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

Step 1.10.2.1.2

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

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Step 1.10.2.1.2.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.10.2.1.2.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 1.10.2.1.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.10.2.1.2.4

Add $1$ and $1$ .

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

Step 1.10.2.2

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

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

Step 1.10.2.3

Rearrange terms.

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

Step 1.10.2.4

Apply pythagorean identity.

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

Step 1.11

Evaluate the derivative at $x = π$ .

$\frac{1 + \sin (π)}{\cos^{2} (π)}$

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.

$\frac{1 + \sin (0)}{\cos^{2} (π)}$

Step 1.12.1.2

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

$\frac{1 + 0}{\cos^{2} (π)}$

Step 1.12.1.3

Add $1$ and $0$ .

$\frac{1}{\cos^{2} (π)}$

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.

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

Step 1.12.2.2

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

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

Step 1.12.2.3

Multiply $- 1$ by $1$ .

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

Step 1.12.2.4

Raise $- 1$ to the power of $2$ .

$\frac{1}{1}$

Step 1.12.3

Cancel the common factor of $1$ .

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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