Calculus Examples

$\sin (3 y) = x$ ; $(0, \frac{π}{3})$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = \frac{π}{3}$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (\sin (3 y)) = \frac{d}{d x} (x)$

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 $f (x) = \sin (x)$ and $g (x) = 3 y$ .

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Step 1.2.1.1

To apply the Chain Rule, set $u$ as $3 y$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [3 y]$

Step 1.2.1.2

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

$\cos (u) \frac{d}{d x} [3 y]$

Step 1.2.1.3

Replace all occurrences of $u$ with $3 y$ .

$\cos (3 y) \frac{d}{d x} [3 y]$

Step 1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 y$ with respect to $x$ is $3 \frac{d}{d x} [y]$ .

$\cos (3 y) (3 \frac{d}{d x} [y])$

Step 1.2.3

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

$\cos (3 y) (3 y')$

Step 1.2.4

Reorder the factors of $\cos (3 y) (3 y')$ .

$3 \cos (3 y) y'$

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.

$3 \cos (3 y) y' = 1$

Step 1.5

Divide each term in $3 \cos (3 y) y' = 1$ by $3 \cos (3 y)$ and simplify.

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Step 1.5.1

Divide each term in $3 \cos (3 y) y' = 1$ by $3 \cos (3 y)$ .

$\frac{3 \cos (3 y) y'}{3 \cos (3 y)} = \frac{1}{3 \cos (3 y)}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{3 \cos (3 y) y'}{3 \cos (3 y)} = \frac{1}{3 \cos (3 y)}$

Step 1.5.2.1.2

Rewrite the expression.

$\frac{\cos (3 y) y'}{\cos (3 y)} = \frac{1}{3 \cos (3 y)}$

Step 1.5.2.2

Cancel the common factor of $\cos (3 y)$ .

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Step 1.5.2.2.1

Cancel the common factor.

$\frac{\cos (3 y) y'}{\cos (3 y)} = \frac{1}{3 \cos (3 y)}$

Step 1.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{1}{3 \cos (3 y)}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Separate fractions.

$y' = \frac{1}{3} \cdot \frac{1}{\cos (3 y)}$

Step 1.5.3.2

Convert from $\frac{1}{\cos (3 y)}$ to $\sec (3 y)$ .

$y' = \frac{1}{3} \sec (3 y)$

Step 1.5.3.3

Combine $\frac{1}{3}$ and $\sec (3 y)$ .

$y' = \frac{\sec (3 y)}{3}$

Step 1.6

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

$\frac{d y}{d x} = \frac{\sec (3 y)}{3}$

Step 1.7

Evaluate at $x = 0$ and $y = \frac{π}{3}$ .

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Step 1.7.1

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

$\frac{\sec (3 y)}{3}$

Step 1.7.2

Replace the variable $y$ with $\frac{π}{3}$ in the expression.

$\frac{\sec (3 (\frac{π}{3}))}{3}$

Step 1.7.3

Simplify the numerator.

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Step 1.7.3.1

Cancel the common factor of $3$ .

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Step 1.7.3.1.1

Cancel the common factor.

$\frac{\sec (3 \frac{π}{3})}{3}$

Step 1.7.3.1.2

Rewrite the expression.

$\frac{\sec (π)}{3}$

Step 1.7.3.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because secant is negative in the second quadrant.

$\frac{- \sec (0)}{3}$

Step 1.7.3.3

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

$\frac{- 1 \cdot 1}{3}$

Step 1.7.3.4

Multiply $- 1$ by $1$ .

$\frac{- 1}{3}$

Step 1.7.4

Move the negative in front of the fraction.

$- \frac{1}{3}$

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 $- \frac{1}{3}$ and a given point $(0, \frac{π}{3})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{π}{3}) = - \frac{1}{3} \cdot (x - (0))$

Step 2.2

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

$y - \frac{π}{3} = - \frac{1}{3} \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{1}{3} \cdot (x + 0)$ .

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Step 2.3.1.1

Add $x$ and $0$ .

$y - \frac{π}{3} = - \frac{1}{3} \cdot x$

Step 2.3.1.2

Combine $x$ and $\frac{1}{3}$ .

$y - \frac{π}{3} = - \frac{x}{3}$

Step 2.3.2

Add $\frac{π}{3}$ to both sides of the equation.

$y = - \frac{x}{3} + \frac{π}{3}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{1}{3} x) + \frac{π}{3}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{1}{3} x + \frac{π}{3}$

Step 3