Calculus Examples

$6 x y + π \sin (y) = 83 π$ , $(4, \frac{7 π}{2})$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = \frac{7 π}{2}$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (6 x y + π \sin (y)) = \frac{d}{d x} (83 π)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

By the Sum Rule, the derivative of $6 x y + π \sin (y)$ with respect to $x$ is $\frac{d}{d x} [6 x y] + \frac{d}{d x} [π \sin (y)]$ .

$\frac{d}{d x} [6 x y] + \frac{d}{d x} [π \sin (y)]$

Step 1.2.2

Evaluate $\frac{d}{d x} [6 x y]$ .

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Step 1.2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x y$ with respect to $x$ is $6 \frac{d}{d x} [x y]$ .

$6 \frac{d}{d x} [x y] + \frac{d}{d x} [π \sin (y)]$

Step 1.2.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = y$ .

$6 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [π \sin (y)]$

Step 1.2.2.3

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

$6 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [π \sin (y)]$

Step 1.2.2.4

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

$6 (x y' + y \cdot 1) + \frac{d}{d x} [π \sin (y)]$

Step 1.2.2.5

Multiply $y$ by $1$ .

$6 (x y' + y) + \frac{d}{d x} [π \sin (y)]$

Step 1.2.3

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

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Step 1.2.3.1

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

$6 (x y' + y) + π \frac{d}{d x} [\sin (y)]$

Step 1.2.3.2

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

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Step 1.2.3.2.1

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

$6 (x y' + y) + π (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [y])$

Step 1.2.3.2.2

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

$6 (x y' + y) + π (\cos (u) \frac{d}{d x} [y])$

Step 1.2.3.2.3

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

$6 (x y' + y) + π (\cos (y) \frac{d}{d x} [y])$

Step 1.2.3.3

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

$6 (x y' + y) + π \cos (y) y'$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Apply the distributive property.

$6 (x y') + 6 y + π \cos (y) y'$

Step 1.2.4.2

Reorder terms.

$6 x y' + π \cos (y) y' + 6 y$

Step 1.3

Since $83 π$ is constant with respect to $x$ , the derivative of $83 π$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

$6 x y' + π \cos (y) y' + 6 y = 0$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Simplify the left side.

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Step 1.5.1.1

Reorder factors in $6 x y' + π \cos (y) y' + 6 y$ .

$6 x y' + π y' \cos (y) + 6 y = 0$

Step 1.5.2

Subtract $6 y$ from both sides of the equation.

$6 x y' + π y' \cos (y) = - 6 y$

Step 1.5.3

Factor $y'$ out of $6 x y' + π y' \cos (y)$ .

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Step 1.5.3.1

Factor $y'$ out of $6 x y'$ .

$y' (6 x) + π y' \cos (y) = - 6 y$

Step 1.5.3.2

Factor $y'$ out of $π y' \cos (y)$ .

$y' (6 x) + y' (π \cos (y)) = - 6 y$

Step 1.5.3.3

Factor $y'$ out of $y' (6 x) + y' (π \cos (y))$ .

$y' (6 x + π \cos (y)) = - 6 y$

Step 1.5.4

Divide each term in $y' (6 x + π \cos (y)) = - 6 y$ by $6 x + π \cos (y)$ and simplify.

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Step 1.5.4.1

Divide each term in $y' (6 x + π \cos (y)) = - 6 y$ by $6 x + π \cos (y)$ .

$\frac{y' (6 x + π \cos (y))}{6 x + π \cos (y)} = \frac{- 6 y}{6 x + π \cos (y)}$

Step 1.5.4.2

Simplify the left side.

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Step 1.5.4.2.1

Cancel the common factor of $6 x + π \cos (y)$ .

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Step 1.5.4.2.1.1

Cancel the common factor.

$\frac{y' (6 x + π \cos (y))}{6 x + π \cos (y)} = \frac{- 6 y}{6 x + π \cos (y)}$

Step 1.5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 6 y}{6 x + π \cos (y)}$

Step 1.5.4.3

Simplify the right side.

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Step 1.5.4.3.1

Move the negative in front of the fraction.

$y' = - \frac{6 y}{6 x + π \cos (y)}$

Step 1.6

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

$\frac{d y}{d x} = - \frac{6 y}{6 x + π \cos (y)}$

Step 1.7

Evaluate at $x = 4$ and $y = \frac{7 π}{2}$ .

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Step 1.7.1

Replace the variable $x$ with $4$ in the expression.

$- \frac{6 y}{6 (4) + π \cos (y)}$

Step 1.7.2

Replace the variable $y$ with $\frac{7 π}{2}$ in the expression.

$- \frac{6 (\frac{7 π}{2})}{6 (4) + π \cos (\frac{7 π}{2})}$

Step 1.7.3

Combine $6$ and $\frac{7 π}{2}$ .

$- \frac{\frac{6 (7 π)}{2}}{6 (4) + π \cos (\frac{7 π}{2})}$

Step 1.7.4

Simplify the denominator.

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Step 1.7.4.1

Multiply $6$ by $4$ .

$- \frac{\frac{6 (7 π)}{2}}{24 + π \cos (\frac{7 π}{2})}$

Step 1.7.4.2

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$- \frac{\frac{6 (7 π)}{2}}{24 + π \cos (\frac{3 π}{2})}$

Step 1.7.4.3

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

$- \frac{\frac{6 (7 π)}{2}}{24 + π \cos (\frac{π}{2})}$

Step 1.7.4.4

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$- \frac{\frac{6 (7 π)}{2}}{24 + π \cdot 0}$

Step 1.7.4.5

Multiply $π$ by $0$ .

$- \frac{\frac{6 (7 π)}{2}}{24 + 0}$

Step 1.7.4.6

Add $24$ and $0$ .

$- \frac{\frac{6 (7 π)}{2}}{24}$

Step 1.7.5

Multiply $6$ by $7$ .

$- \frac{\frac{42 π}{2}}{24}$

Step 1.7.6

Reduce the expression by cancelling the common factors.

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Step 1.7.6.1

Reduce the expression $\frac{42 π}{2}$ by cancelling the common factors.

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Step 1.7.6.1.1

Factor $2$ out of $42 π$ .

$- \frac{\frac{2 (21 π)}{2}}{24}$

Step 1.7.6.1.2

Factor $2$ out of $2$ .

$- \frac{\frac{2 (21 π)}{2 (1)}}{24}$

Step 1.7.6.1.3

Cancel the common factor.

$- \frac{\frac{2 (21 π)}{2 \cdot 1}}{24}$

Step 1.7.6.1.4

Rewrite the expression.

$- \frac{\frac{21 π}{1}}{24}$

Step 1.7.6.2

Divide $21 π$ by $1$ .

$- \frac{21 π}{24}$

Step 1.7.7

Cancel the common factor of $21$ and $24$ .

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Step 1.7.7.1

Factor $3$ out of $21 π$ .

$- \frac{3 (7 π)}{24}$

Step 1.7.7.2

Cancel the common factors.

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Step 1.7.7.2.1

Factor $3$ out of $24$ .

$- \frac{3 (7 π)}{3 (8)}$

Step 1.7.7.2.2

Cancel the common factor.

$- \frac{3 (7 π)}{3 \cdot 8}$

Step 1.7.7.2.3

Rewrite the expression.

$- \frac{7 π}{8}$

Step 2

The normal line is perpendicular to the tangent line. Take the negative reciprocal of the slope of the tangent line to find the slope of the normal line.

$\frac{8}{7 π}$

Step 3

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 3.1

Use the slope $\frac{8}{7 π}$ and a given point $(4, \frac{7 π}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{7 π}{2}) = \frac{8}{7 π} \cdot (x - (4))$

Step 3.2

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

$y - \frac{7 π}{2} = \frac{8}{7 π} \cdot (x - 4)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $\frac{8}{7 π} \cdot (x - 4)$ .

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Step 3.3.1.1

Rewrite.

$y - \frac{7 π}{2} = 0 + 0 + \frac{8}{7 π} \cdot (x - 4)$

Step 3.3.1.2

Simplify by adding zeros.

$y - \frac{7 π}{2} = \frac{8}{7 π} \cdot (x - 4)$

Step 3.3.1.3

Apply the distributive property.

$y - \frac{7 π}{2} = \frac{8}{7 π} x + \frac{8}{7 π} \cdot - 4$

Step 3.3.1.4

Combine $\frac{8}{7 π}$ and $x$ .

$y - \frac{7 π}{2} = \frac{8 x}{7 π} + \frac{8}{7 π} \cdot - 4$

Step 3.3.1.5

Multiply $\frac{8}{7 π} \cdot - 4$ .

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Step 3.3.1.5.1

Combine $\frac{8}{7 π}$ and $- 4$ .

$y - \frac{7 π}{2} = \frac{8 x}{7 π} + \frac{8 \cdot - 4}{7 π}$

Step 3.3.1.5.2

Multiply $8$ by $- 4$ .

$y - \frac{7 π}{2} = \frac{8 x}{7 π} + \frac{- 32}{7 π}$

Step 3.3.1.6

Move the negative in front of the fraction.

$y - \frac{7 π}{2} = \frac{8 x}{7 π} - \frac{32}{7 π}$

Step 3.3.2

Add $\frac{7 π}{2}$ to both sides of the equation.

$y = \frac{8 x}{7 π} - \frac{32}{7 π} + \frac{7 π}{2}$

Step 3.3.3

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

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Step 3.3.3.1

To write $- \frac{32}{7 π}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{8 x}{7 π} - \frac{32}{7 π} \cdot \frac{2}{2} + \frac{7 π}{2}$

Step 3.3.3.2

To write $\frac{7 π}{2}$ as a fraction with a common denominator, multiply by $\frac{7 π}{7 π}$ .

$y = \frac{8 x}{7 π} - \frac{32}{7 π} \cdot \frac{2}{2} + \frac{7 π}{2} \cdot \frac{7 π}{7 π}$

Step 3.3.3.3

Write each expression with a common denominator of $14 π$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.3.3.1

Multiply $\frac{32}{7 π}$ by $\frac{2}{2}$ .

$y = \frac{8 x}{7 π} - \frac{32 \cdot 2}{7 π \cdot 2} + \frac{7 π}{2} \cdot \frac{7 π}{7 π}$

Step 3.3.3.3.2

Multiply $2$ by $7$ .

$y = \frac{8 x}{7 π} - \frac{32 \cdot 2}{14 π} + \frac{7 π}{2} \cdot \frac{7 π}{7 π}$

Step 3.3.3.3.3

Multiply $\frac{7 π}{2}$ by $\frac{7 π}{7 π}$ .

$y = \frac{8 x}{7 π} - \frac{32 \cdot 2}{14 π} + \frac{7 π (7 π)}{2 (7 π)}$

Step 3.3.3.3.4

Multiply $7$ by $2$ .

$y = \frac{8 x}{7 π} - \frac{32 \cdot 2}{14 π} + \frac{7 π (7 π)}{14 π}$

Step 3.3.3.4

Combine the numerators over the common denominator.

$y = \frac{8 x}{7 π} + \frac{- 32 \cdot 2 + 7 π (7 π)}{14 π}$

Step 3.3.3.5

Simplify the numerator.

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Step 3.3.3.5.1

Multiply $- 32$ by $2$ .

$y = \frac{8 x}{7 π} + \frac{- 64 + 7 π \cdot 7 π}{14 π}$

Step 3.3.3.5.2

Multiply $7$ by $7$ .

$y = \frac{8 x}{7 π} + \frac{- 64 + 49 π π}{14 π}$

Step 3.3.3.5.3

Multiply $49 π π$ .

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Step 3.3.3.5.3.1

Raise $π$ to the power of $1$ .

$y = \frac{8 x}{7 π} + \frac{- 64 + 49 (π^{1} π)}{14 π}$

Step 3.3.3.5.3.2

Raise $π$ to the power of $1$ .

$y = \frac{8 x}{7 π} + \frac{- 64 + 49 (π^{1} π^{1})}{14 π}$

Step 3.3.3.5.3.3

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

$y = \frac{8 x}{7 π} + \frac{- 64 + 49 π^{1 + 1}}{14 π}$

Step 3.3.3.5.3.4

Add $1$ and $1$ .

$y = \frac{8 x}{7 π} + \frac{- 64 + 49 π^{2}}{14 π}$

Step 3.3.3.5.4

Rewrite $- 64 + 49 π^{2}$ in a factored form.

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Step 3.3.3.5.4.1

Rewrite $49 π^{2}$ as ${(7 π)}^{2}$ .

$y = \frac{8 x}{7 π} + \frac{- 64 + {(7 π)}^{2}}{14 π}$

Step 3.3.3.5.4.2

Rewrite $64$ as $8^{2}$ .

$y = \frac{8 x}{7 π} + \frac{- 8^{2} + {(7 π)}^{2}}{14 π}$

Step 3.3.3.5.4.3

Reorder $- 8^{2}$ and ${(7 π)}^{2}$ .

$y = \frac{8 x}{7 π} + \frac{{(7 π)}^{2} - 8^{2}}{14 π}$

Step 3.3.3.5.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7 π$ and $b = 8$ .

$y = \frac{8 x}{7 π} + \frac{(7 π + 8) (7 π - 8)}{14 π}$

Step 3.3.3.6

Reorder terms.

$y = \frac{8}{7 π} x + \frac{(7 π + 8) (7 π - 8)}{14 π}$

Step 4