Calculus Examples

$y = \frac{8}{\sin (x) + \cos (x)}$ , $(0, 8)$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = 8$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{\sin (x) + \cos (x)}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{\sin (x) + \cos (x)}]$ .

$8 \frac{d}{d x} [\frac{1}{\sin (x) + \cos (x)}]$

Step 1.1.2

Rewrite $\frac{1}{\sin (x) + \cos (x)}$ as ${(\sin (x) + \cos (x))}^{- 1}$ .

$8 \frac{d}{d x} [{(\sin (x) + \cos (x))}^{- 1}]$

Step 1.2

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

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\sin (x) + \cos (x)$ .

$8 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [\sin (x) + \cos (x)])$

Step 1.2.2

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

$8 (- u^{- 2} \frac{d}{d x} [\sin (x) + \cos (x)])$

Step 1.2.3

Replace all occurrences of $u$ with $\sin (x) + \cos (x)$ .

$8 (- {(\sin (x) + \cos (x))}^{- 2} \frac{d}{d x} [\sin (x) + \cos (x)])$

Step 1.3

Differentiate using the Sum Rule.

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Step 1.3.1

Multiply $- 1$ by $8$ .

$- 8 ({(\sin (x) + \cos (x))}^{- 2} \frac{d}{d x} [\sin (x) + \cos (x)])$

Step 1.3.2

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

$- 8 {(\sin (x) + \cos (x))}^{- 2} (\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)])$

Step 1.4

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

$- 8 {(\sin (x) + \cos (x))}^{- 2} (\cos (x) + \frac{d}{d x} [\cos (x)])$

Step 1.5

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

$- 8 {(\sin (x) + \cos (x))}^{- 2} (\cos (x) - \sin (x))$

Step 1.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 8 \frac{1}{{(\sin (x) + \cos (x))}^{2}} (\cos (x) - \sin (x))$

Step 1.7

Simplify.

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Step 1.7.1

Combine terms.

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Step 1.7.1.1

Combine $- 8$ and $\frac{1}{{(\sin (x) + \cos (x))}^{2}}$ .

$\frac{- 8}{{(\sin (x) + \cos (x))}^{2}} (\cos (x) - \sin (x))$

Step 1.7.1.2

Move the negative in front of the fraction.

$- \frac{8}{{(\sin (x) + \cos (x))}^{2}} (\cos (x) - \sin (x))$

Step 1.7.2

Reorder the factors of $- \frac{8}{{(\sin (x) + \cos (x))}^{2}} (\cos (x) - \sin (x))$ .

$- (\cos (x) - \sin (x)) \frac{8}{{(\sin (x) + \cos (x))}^{2}}$

Step 1.8

Evaluate the derivative at $x = 0$ .

$- (\cos (0) - \sin (0)) \frac{8}{{(\sin (0) + \cos (0))}^{2}}$

Step 1.9

Simplify.

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Step 1.9.1

Simplify the denominator.

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Step 1.9.1.1

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

$- (\cos (0) - \sin (0)) \frac{8}{{(0 + \cos (0))}^{2}}$

Step 1.9.1.2

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

$- (\cos (0) - \sin (0)) \frac{8}{{(0 + 1)}^{2}}$

Step 1.9.1.3

Add $0$ and $1$ .

$- (\cos (0) - \sin (0)) \frac{8}{1^{2}}$

Step 1.9.1.4

One to any power is one.

$- (\cos (0) - \sin (0)) \frac{8}{1}$

Step 1.9.2

Simplify terms.

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Step 1.9.2.1

Simplify each term.

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Step 1.9.2.1.1

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

$- (1 - \sin (0)) \frac{8}{1}$

Step 1.9.2.1.2

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

$- (1 - 0) \frac{8}{1}$

Step 1.9.2.1.3

Multiply $- 1$ by $0$ .

$- (1 + 0) \frac{8}{1}$

Step 1.9.2.2

Simplify the expression.

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Step 1.9.2.2.1

Add $1$ and $0$ .

$- 1 \cdot 1 \frac{8}{1}$

Step 1.9.2.2.2

Multiply $- 1$ by $1$ .

$- 1 (\frac{8}{1})$

Step 1.9.2.2.3

Divide $8$ by $1$ .

$- 1 \cdot 8$

Step 1.9.2.2.4

Multiply $- 1$ by $8$ .

$- 8$

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 $(0, 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (8) = - 8 \cdot (x - (0))$

Step 2.2

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

$y - 8 = - 8 \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $x$ and $0$ .

$y - 8 = - 8 x$

Step 2.3.2

Add $8$ to both sides of the equation.

$y = - 8 x + 8$

Step 3