Calculus Examples

$\cos (4 y) = x$ , $(0, \frac{π}{8})$

Step 1

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

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (\cos (4 y)) = \frac{d}{d x} (x)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 4 y$ .

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Step 1.2.1.1

To apply the Chain Rule, set $u$ as $4 y$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d x} [4 y]$

Step 1.2.1.2

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

$- \sin (u) \frac{d}{d x} [4 y]$

Step 1.2.1.3

Replace all occurrences of $u$ with $4 y$ .

$- \sin (4 y) \frac{d}{d x} [4 y]$

Step 1.2.2

Differentiate using the Constant Multiple Rule.

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Step 1.2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 y$ with respect to $x$ is $4 \frac{d}{d x} [y]$ .

$- \sin (4 y) (4 \frac{d}{d x} [y])$

Step 1.2.2.2

Multiply $4$ by $- 1$ .

$- 4 \sin (4 y) \frac{d}{d x} [y]$

Step 1.2.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$- 4 \sin (4 y) y'$

Step 1.3

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

$1$

Step 1.4

Reform the equation by setting the left side equal to the right side.

$- 4 \sin (4 y) y' = 1$

Step 1.5

Divide each term in $- 4 \sin (4 y) y' = 1$ by $- 4 \sin (4 y)$ and simplify.

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Step 1.5.1

Divide each term in $- 4 \sin (4 y) y' = 1$ by $- 4 \sin (4 y)$ .

$\frac{- 4 \sin (4 y) y'}{- 4 \sin (4 y)} = \frac{1}{- 4 \sin (4 y)}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $- 4$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{- 4 \sin (4 y) y'}{- 4 \sin (4 y)} = \frac{1}{- 4 \sin (4 y)}$

Step 1.5.2.1.2

Rewrite the expression.

$\frac{\sin (4 y) y'}{\sin (4 y)} = \frac{1}{- 4 \sin (4 y)}$

Step 1.5.2.2

Cancel the common factor of $\sin (4 y)$ .

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Step 1.5.2.2.1

Cancel the common factor.

$\frac{\sin (4 y) y'}{\sin (4 y)} = \frac{1}{- 4 \sin (4 y)}$

Step 1.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{1}{- 4 \sin (4 y)}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Separate fractions.

$y' = \frac{1}{- 4} \cdot \frac{1}{\sin (4 y)}$

Step 1.5.3.2

Convert from $\frac{1}{\sin (4 y)}$ to $\csc (4 y)$ .

$y' = \frac{1}{- 4} \csc (4 y)$

Step 1.5.3.3

Move the negative in front of the fraction.

$y' = - \frac{1}{4} \csc (4 y)$

Step 1.5.3.4

Combine $\csc (4 y)$ and $\frac{1}{4}$ .

$y' = - \frac{\csc (4 y)}{4}$

Step 1.6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{\csc (4 y)}{4}$

Step 1.7

Evaluate at $x = 0$ and $y = \frac{π}{8}$ .

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Step 1.7.1

Replace the variable $x$ with $0$ in the expression.

$- \frac{\csc (4 y)}{4}$

Step 1.7.2

Replace the variable $y$ with $\frac{π}{8}$ in the expression.

$- \frac{\csc (4 (\frac{π}{8}))}{4}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

Cancel the common factor of $4$ .

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Step 1.7.3.1.1

Factor $4$ out of $8$ .

$- \frac{\csc (4 \frac{π}{4 (2)})}{4}$

Step 1.7.3.1.2

Cancel the common factor.

$- \frac{\csc (4 \frac{π}{4 \cdot 2})}{4}$

Step 1.7.3.1.3

Rewrite the expression.

$- \frac{\csc (\frac{π}{2})}{4}$

Step 1.7.3.2

The exact value of $\csc (\frac{π}{2})$ is $1$ .

$- \frac{1}{4}$

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 $- \frac{1}{4}$ and a given point $(0, \frac{π}{8})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{π}{8}) = - \frac{1}{4} \cdot (x - (0))$

Step 2.2

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

$y - \frac{π}{8} = - \frac{1}{4} \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{1}{4} \cdot (x + 0)$ .

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Step 2.3.1.1

Add $x$ and $0$ .

$y - \frac{π}{8} = - \frac{1}{4} \cdot x$

Step 2.3.1.2

Combine $x$ and $\frac{1}{4}$ .

$y - \frac{π}{8} = - \frac{x}{4}$

Step 2.3.2

Add $\frac{π}{8}$ to both sides of the equation.

$y = - \frac{x}{4} + \frac{π}{8}$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{1}{4} x) + \frac{π}{8}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{1}{4} x + \frac{π}{8}$

Step 3