Calculus Examples

$8 x - y^{2} = 23 y$ ; $(3, 1)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (8 x - y^{2}) = \frac{d}{d x} (23 y)$

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 $8 x - y^{2}$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [- y^{2}]$ .

$\frac{d}{d x} [8 x] + \frac{d}{d x} [- y^{2}]$

Step 1.2.2

Evaluate $\frac{d}{d x} [8 x]$ .

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Step 1.2.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x] + \frac{d}{d x} [- y^{2}]$

Step 1.2.2.2

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

$8 \cdot 1 + \frac{d}{d x} [- y^{2}]$

Step 1.2.2.3

Multiply $8$ by $1$ .

$8 + \frac{d}{d x} [- y^{2}]$

Step 1.2.3

Evaluate $\frac{d}{d x} [- y^{2}]$ .

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Step 1.2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- y^{2}$ with respect to $x$ is $- \frac{d}{d x} [y^{2}]$ .

$8 - \frac{d}{d x} [y^{2}]$

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 $f (x) = x^{2}$ and $g (x) = y$ .

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Step 1.2.3.2.1

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

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

Step 1.2.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$8 - (2 u \frac{d}{d x} [y])$

Step 1.2.3.2.3

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

$8 - (2 y \frac{d}{d x} [y])$

Step 1.2.3.3

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

$8 - (2 y y')$

Step 1.2.3.4

Multiply $2$ by $- 1$ .

$8 - 2 y y'$

Step 1.2.4

Reorder terms.

$- 2 y y' + 8$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

Since $23$ is constant with respect to $x$ , the derivative of $23 y$ with respect to $x$ is $23 \frac{d}{d x} [y]$ .

$23 \frac{d}{d x} [y]$

Step 1.3.2

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

$23 y'$

Step 1.4

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

$- 2 y y' + 8 = 23 y'$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Subtract $23 y'$ from both sides of the equation.

$- 2 y y' + 8 - 23 y' = 0$

Step 1.5.2

Subtract $8$ from both sides of the equation.

$- 2 y y' - 23 y' = - 8$

Step 1.5.3

Factor $y'$ out of $- 2 y y' - 23 y'$ .

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Step 1.5.3.1

Factor $y'$ out of $- 2 y y'$ .

$y' (- 2 y) - 23 y' = - 8$

Step 1.5.3.2

Factor $y'$ out of $- 23 y'$ .

$y' (- 2 y) + y' \cdot - 23 = - 8$

Step 1.5.3.3

Factor $y'$ out of $y' (- 2 y) + y' \cdot - 23$ .

$y' (- 2 y - 23) = - 8$

Step 1.5.4

Divide each term in $y' (- 2 y - 23) = - 8$ by $- 2 y - 23$ and simplify.

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Step 1.5.4.1

Divide each term in $y' (- 2 y - 23) = - 8$ by $- 2 y - 23$ .

$\frac{y' (- 2 y - 23)}{- 2 y - 23} = \frac{- 8}{- 2 y - 23}$

Step 1.5.4.2

Simplify the left side.

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Step 1.5.4.2.1

Cancel the common factor of $- 2 y - 23$ .

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Step 1.5.4.2.1.1

Cancel the common factor.

$\frac{y' (- 2 y - 23)}{- 2 y - 23} = \frac{- 8}{- 2 y - 23}$

Step 1.5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 8}{- 2 y - 23}$

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.

$y' = - \frac{8}{- 2 y - 23}$

Step 1.5.4.3.2

Factor $- 1$ out of $- 2 y$ .

$y' = - \frac{8}{- (2 y) - 23}$

Step 1.5.4.3.3

Rewrite $- 23$ as $- 1 (23)$ .

$y' = - \frac{8}{- (2 y) - 1 (23)}$

Step 1.5.4.3.4

Factor $- 1$ out of $- (2 y) - 1 (23)$ .

$y' = - \frac{8}{- (2 y + 23)}$

Step 1.5.4.3.5

Simplify the expression.

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Step 1.5.4.3.5.1

Rewrite $- (2 y + 23)$ as $- 1 (2 y + 23)$ .

$y' = - \frac{8}{- 1 (2 y + 23)}$

Step 1.5.4.3.5.2

Move the negative in front of the fraction.

$y' = - - \frac{8}{2 y + 23}$

Step 1.5.4.3.5.3

Multiply $- 1$ by $- 1$ .

$y' = 1 \frac{8}{2 y + 23}$

Step 1.5.4.3.5.4

Multiply $\frac{8}{2 y + 23}$ by $1$ .

$y' = \frac{8}{2 y + 23}$

Step 1.6

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

$\frac{d y}{d x} = \frac{8}{2 y + 23}$

Step 1.7

Evaluate at $x = 3$ and $y = 1$ .

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Step 1.7.1

Replace the variable $x$ with $3$ in the expression.

$\frac{8}{2 y + 23}$

Step 1.7.2

Replace the variable $y$ with $1$ in the expression.

$\frac{8}{2 (1) + 23}$

Step 1.7.3

Simplify the denominator.

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Step 1.7.3.1

Multiply $2$ by $1$ .

$\frac{8}{2 + 23}$

Step 1.7.3.2

Add $2$ and $23$ .

$\frac{8}{25}$

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 $\frac{8}{25}$ and a given point $(3, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = \frac{8}{25} \cdot (x - (3))$

Step 2.2

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

$y - 1 = \frac{8}{25} \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{8}{25} \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 + \frac{8}{25} \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = \frac{8}{25} \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = \frac{8}{25} x + \frac{8}{25} \cdot - 3$

Step 2.3.1.4

Combine $\frac{8}{25}$ and $x$ .

$y - 1 = \frac{8 x}{25} + \frac{8}{25} \cdot - 3$

Step 2.3.1.5

Multiply $\frac{8}{25} \cdot - 3$ .

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Step 2.3.1.5.1

Combine $\frac{8}{25}$ and $- 3$ .

$y - 1 = \frac{8 x}{25} + \frac{8 \cdot - 3}{25}$

Step 2.3.1.5.2

Multiply $8$ by $- 3$ .

$y - 1 = \frac{8 x}{25} + \frac{- 24}{25}$

Step 2.3.1.6

Move the negative in front of the fraction.

$y - 1 = \frac{8 x}{25} - \frac{24}{25}$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.3.2.1

Add $1$ to both sides of the equation.

$y = \frac{8 x}{25} - \frac{24}{25} + 1$

Step 2.3.2.2

Write $1$ as a fraction with a common denominator.

$y = \frac{8 x}{25} - \frac{24}{25} + \frac{25}{25}$

Step 2.3.2.3

Combine the numerators over the common denominator.

$y = \frac{8 x}{25} + \frac{- 24 + 25}{25}$

Step 2.3.2.4

Add $- 24$ and $25$ .

$y = \frac{8 x}{25} + \frac{1}{25}$

Step 2.3.3

Reorder terms.

$y = \frac{8}{25} x + \frac{1}{25}$

Step 3