Calculus Examples

$x^{2} y^{8} = 9$ , $(3, 1)$

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.

$\frac{d}{d x} (x^{2} y^{8}) = \frac{d}{d x} (9)$

Step 1.2

Differentiate the left side of the equation.

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Step 1.2.1

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 $f (x) = x^{2}$ and $g (x) = y^{8}$ .

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

Step 1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{8}$ and $g (x) = y$ .

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Step 1.2.2.1

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

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

Step 1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 8$ .

$x^{2} (8 u^{7} \frac{d}{d x} [y]) + y^{8} \frac{d}{d x} [x^{2}]$

Step 1.2.2.3

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

$x^{2} (8 y^{7} \frac{d}{d x} [y]) + y^{8} \frac{d}{d x} [x^{2}]$

Step 1.2.3

Move $8$ to the left of $x^{2}$ .

$8 \cdot x^{2} y^{7} \frac{d}{d x} [y] + y^{8} \frac{d}{d x} [x^{2}]$

Step 1.2.4

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

$8 x^{2} y^{7} y' + y^{8} \frac{d}{d x} [x^{2}]$

Step 1.2.5

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

$8 x^{2} y^{7} y' + y^{8} (2 x)$

Step 1.2.6

Reorder.

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Step 1.2.6.1

Move $2$ to the left of $y^{8}$ .

$8 x^{2} y^{7} y' + 2 \cdot y^{8} x$

Step 1.2.6.2

Reorder terms.

$8 y^{7} x^{2} y' + 2 y^{8} x$

Step 1.3

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

$8 y^{7} x^{2} y' + 2 y^{8} x = 0$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Subtract $2 y^{8} x$ from both sides of the equation.

$8 y^{7} x^{2} y' = - 2 y^{8} x$

Step 1.5.2

Divide each term in $8 y^{7} x^{2} y' = - 2 y^{8} x$ by $8 y^{7} x^{2}$ and simplify.

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Step 1.5.2.1

Divide each term in $8 y^{7} x^{2} y' = - 2 y^{8} x$ by $8 y^{7} x^{2}$ .

$\frac{8 y^{7} x^{2} y'}{8 y^{7} x^{2}} = \frac{- 2 y^{8} x}{8 y^{7} x^{2}}$

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $8$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

$\frac{8 y^{7} x^{2} y'}{8 y^{7} x^{2}} = \frac{- 2 y^{8} x}{8 y^{7} x^{2}}$

Step 1.5.2.2.1.2

Rewrite the expression.

$\frac{y^{7} x^{2} y'}{y^{7} x^{2}} = \frac{- 2 y^{8} x}{8 y^{7} x^{2}}$

Step 1.5.2.2.2

Cancel the common factor of $y^{7}$ .

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Step 1.5.2.2.2.1

Cancel the common factor.

$\frac{y^{7} x^{2} y'}{y^{7} x^{2}} = \frac{- 2 y^{8} x}{8 y^{7} x^{2}}$

Step 1.5.2.2.2.2

Rewrite the expression.

$\frac{x^{2} y'}{x^{2}} = \frac{- 2 y^{8} x}{8 y^{7} x^{2}}$

Step 1.5.2.2.3

Cancel the common factor of $x^{2}$ .

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Step 1.5.2.2.3.1

Cancel the common factor.

$\frac{x^{2} y'}{x^{2}} = \frac{- 2 y^{8} x}{8 y^{7} x^{2}}$

Step 1.5.2.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 y^{8} x}{8 y^{7} x^{2}}$

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 $8$ .

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Step 1.5.2.3.1.1

Factor $2$ out of $- 2 y^{8} x$ .

$y' = \frac{2 (- y^{8} x)}{8 y^{7} x^{2}}$

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 $8 y^{7} x^{2}$ .

$y' = \frac{2 (- y^{8} x)}{2 (4 y^{7} x^{2})}$

Step 1.5.2.3.1.2.2

Cancel the common factor.

$y' = \frac{2 (- y^{8} x)}{2 (4 y^{7} x^{2})}$

Step 1.5.2.3.1.2.3

Rewrite the expression.

$y' = \frac{- y^{8} x}{4 y^{7} x^{2}}$

Step 1.5.2.3.2

Cancel the common factor of $y^{8}$ and $y^{7}$ .

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Step 1.5.2.3.2.1

Factor $y^{7}$ out of $- y^{8} x$ .

$y' = \frac{y^{7} (- y x)}{4 y^{7} x^{2}}$

Step 1.5.2.3.2.2

Cancel the common factors.

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Step 1.5.2.3.2.2.1

Factor $y^{7}$ out of $4 y^{7} x^{2}$ .

$y' = \frac{y^{7} (- y x)}{y^{7} (4 x^{2})}$

Step 1.5.2.3.2.2.2

Cancel the common factor.

$y' = \frac{y^{7} (- y x)}{y^{7} (4 x^{2})}$

Step 1.5.2.3.2.2.3

Rewrite the expression.

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

Step 1.5.2.3.3

Cancel the common factor of $x$ and $x^{2}$ .

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Step 1.5.2.3.3.1

Factor $x$ out of $- y x$ .

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

Step 1.5.2.3.3.2

Cancel the common factors.

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Step 1.5.2.3.3.2.1

Factor $x$ out of $4 x^{2}$ .

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

Step 1.5.2.3.3.2.2

Cancel the common factor.

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

Step 1.5.2.3.3.2.3

Rewrite the expression.

$y' = \frac{- y}{4 x}$

Step 1.5.2.3.4

Move the negative in front of the fraction.

$y' = - \frac{y}{4 x}$

Step 1.6

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

$\frac{d y}{d x} = - \frac{y}{4 x}$

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.

$- \frac{y}{4 (3)}$

Step 1.7.2

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

$- \frac{1}{4 (3)}$

Step 1.7.3

Multiply $4$ by $3$ .

$- \frac{1}{12}$

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 $- \frac{1}{12}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{1}{12} \cdot (x - (3))$

Step 2.2

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

$y - 1 = - \frac{1}{12} \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{1}{12} \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{1}{12} \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = - \frac{1}{12} \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = - \frac{1}{12} x - \frac{1}{12} \cdot - 3$

Step 2.3.1.4

Combine $x$ and $\frac{1}{12}$ .

$y - 1 = - \frac{x}{12} - \frac{1}{12} \cdot - 3$

Step 2.3.1.5

Cancel the common factor of $3$ .

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Step 2.3.1.5.1

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$y - 1 = - \frac{x}{12} + \frac{- 1}{12} \cdot - 3$

Step 2.3.1.5.2

Factor $3$ out of $12$ .

$y - 1 = - \frac{x}{12} + \frac{- 1}{3 (4)} \cdot - 3$

Step 2.3.1.5.3

Factor $3$ out of $- 3$ .

$y - 1 = - \frac{x}{12} + \frac{- 1}{3 \cdot 4} \cdot (3 \cdot - 1)$

Step 2.3.1.5.4

Cancel the common factor.

$y - 1 = - \frac{x}{12} + \frac{- 1}{3 \cdot 4} \cdot (3 \cdot - 1)$

Step 2.3.1.5.5

Rewrite the expression.

$y - 1 = - \frac{x}{12} + \frac{- 1}{4} \cdot - 1$

Step 2.3.1.6

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

$y - 1 = - \frac{x}{12} + \frac{- - 1}{4}$

Step 2.3.1.7

Multiply $- 1$ by $- 1$ .

$y - 1 = - \frac{x}{12} + \frac{1}{4}$

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.

$y = - \frac{x}{12} + \frac{1}{4} + 1$

Step 2.3.2.2

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

$y = - \frac{x}{12} + \frac{1}{4} + \frac{4}{4}$

Step 2.3.2.3

Combine the numerators over the common denominator.

$y = - \frac{x}{12} + \frac{1 + 4}{4}$

Step 2.3.2.4

Add $1$ and $4$ .

$y = - \frac{x}{12} + \frac{5}{4}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{1}{12} x) + \frac{5}{4}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{1}{12} x + \frac{5}{4}$

Step 3