Calculus Examples

Given: $x^{2} = y^{4}$ Find the equation of the tangent line at $(9, 3)$

Step 1

Find the first derivative and evaluate at $x = 9$ and $y = 3$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2}) = \frac{d}{d x} (y^{4})$

Step 1.2

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

$2 x$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

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

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Step 1.3.1.1

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

$\frac{d}{d u} [u^{4}] \frac{d}{d x} [y]$

Step 1.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$4 u^{3} \frac{d}{d x} [y]$

Step 1.3.1.3

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

$4 y^{3} \frac{d}{d x} [y]$

Step 1.3.2

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

$4 y^{3} y'$

Step 1.4

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

$2 x = 4 y^{3} y'$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Rewrite the equation as $4 y^{3} y' = 2 x$ .

$4 y^{3} y' = 2 x$

Step 1.5.2

Divide each term in $4 y^{3} y' = 2 x$ by $4 y^{3}$ and simplify.

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Step 1.5.2.1

Divide each term in $4 y^{3} y' = 2 x$ by $4 y^{3}$ .

$\frac{4 y^{3} y'}{4 y^{3}} = \frac{2 x}{4 y^{3}}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $4$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{4 y^{3} y'}{4 y^{3}} = \frac{2 x}{4 y^{3}}$

Step 1.5.2.2.1.2

Rewrite the expression.

$\frac{y^{3} y'}{y^{3}} = \frac{2 x}{4 y^{3}}$

Step 1.5.2.2.2

Cancel the common factor of $y^{3}$ .

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Step 1.5.2.2.2.1

Cancel the common factor.

$\frac{y^{3} y'}{y^{3}} = \frac{2 x}{4 y^{3}}$

Step 1.5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{2 x}{4 y^{3}}$

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Cancel the common factor of $2$ and $4$ .

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Step 1.5.2.3.1.1

Factor $2$ out of $2 x$ .

$y' = \frac{2 (x)}{4 y^{3}}$

Step 1.5.2.3.1.2

Cancel the common factors.

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Step 1.5.2.3.1.2.1

Factor $2$ out of $4 y^{3}$ .

$y' = \frac{2 (x)}{2 (2 y^{3})}$

Step 1.5.2.3.1.2.2

Cancel the common factor.

$y' = \frac{2 x}{2 (2 y^{3})}$

Step 1.5.2.3.1.2.3

Rewrite the expression.

$y' = \frac{x}{2 y^{3}}$

Step 1.6

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

$\frac{d y}{d x} = \frac{x}{2 y^{3}}$

Step 1.7

Evaluate at $x = 9$ and $y = 3$ .

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Step 1.7.1

Replace the variable $x$ with $9$ in the expression.

$\frac{9}{2 y^{3}}$

Step 1.7.2

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

$\frac{9}{2 {(3)}^{3}}$

Step 1.7.3

Simplify the expression.

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Step 1.7.3.1

Raise $3$ to the power of $3$ .

$\frac{9}{2 \cdot 27}$

Step 1.7.3.2

Multiply $2$ by $27$ .

$\frac{9}{54}$

Step 1.7.4

Cancel the common factor of $9$ and $54$ .

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Step 1.7.4.1

Factor $9$ out of $9$ .

$\frac{9 (1)}{54}$

Step 1.7.4.2

Cancel the common factors.

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Step 1.7.4.2.1

Factor $9$ out of $54$ .

$\frac{9 \cdot 1}{9 \cdot 6}$

Step 1.7.4.2.2

Cancel the common factor.

$\frac{9 \cdot 1}{9 \cdot 6}$

Step 1.7.4.2.3

Rewrite the expression.

$\frac{1}{6}$

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 $\frac{1}{6}$ and a given point $(9, 3)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = \frac{1}{6} \cdot (x - (9))$

Step 2.2

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

$y - 3 = \frac{1}{6} \cdot (x - 9)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{6} \cdot (x - 9)$ .

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Step 2.3.1.1

Rewrite.

$y - 3 = 0 + 0 + \frac{1}{6} \cdot (x - 9)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 3 = \frac{1}{6} \cdot (x - 9)$

Step 2.3.1.3

Apply the distributive property.

$y - 3 = \frac{1}{6} x + \frac{1}{6} \cdot - 9$

Step 2.3.1.4

Combine $\frac{1}{6}$ and $x$ .

$y - 3 = \frac{x}{6} + \frac{1}{6} \cdot - 9$

Step 2.3.1.5

Cancel the common factor of $3$ .

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Step 2.3.1.5.1

Factor $3$ out of $6$ .

$y - 3 = \frac{x}{6} + \frac{1}{3 (2)} \cdot - 9$

Step 2.3.1.5.2

Factor $3$ out of $- 9$ .

$y - 3 = \frac{x}{6} + \frac{1}{3 \cdot 2} \cdot (3 \cdot - 3)$

Step 2.3.1.5.3

Cancel the common factor.

$y - 3 = \frac{x}{6} + \frac{1}{3 \cdot 2} \cdot (3 \cdot - 3)$

Step 2.3.1.5.4

Rewrite the expression.

$y - 3 = \frac{x}{6} + \frac{1}{2} \cdot - 3$

Step 2.3.1.6

Combine $\frac{1}{2}$ and $- 3$ .

$y - 3 = \frac{x}{6} + \frac{- 3}{2}$

Step 2.3.1.7

Move the negative in front of the fraction.

$y - 3 = \frac{x}{6} - \frac{3}{2}$

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 $3$ to both sides of the equation.

$y = \frac{x}{6} - \frac{3}{2} + 3$

Step 2.3.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{x}{6} - \frac{3}{2} + 3 \cdot \frac{2}{2}$

Step 2.3.2.3

Combine $3$ and $\frac{2}{2}$ .

$y = \frac{x}{6} - \frac{3}{2} + \frac{3 \cdot 2}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{6} + \frac{- 3 + 3 \cdot 2}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $3$ by $2$ .

$y = \frac{x}{6} + \frac{- 3 + 6}{2}$

Step 2.3.2.5.2

Add $- 3$ and $6$ .

$y = \frac{x}{6} + \frac{3}{2}$

Step 2.3.3

Reorder terms.

$y = \frac{1}{6} x + \frac{3}{2}$

Step 3