Calculus Examples

$y^{2} = 5 x^{4} - x^{2}$ , $(- 1, 2)$

Step 1

Find the first derivative and evaluate at $x = - 1$ and $y = 2$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2}) = \frac{d}{d x} (5 x^{4} - x^{2})$

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

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

$2 u \frac{d}{d x} [y]$

Step 1.2.1.3

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

$2 y \frac{d}{d x} [y]$

Step 1.2.2

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

$2 y y'$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

By the Sum Rule, the derivative of $5 x^{4} - x^{2}$ with respect to $x$ is $\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- x^{2}]$

Step 1.3.2

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

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Step 1.3.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- x^{2}]$

Step 1.3.2.2

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

$5 (4 x^{3}) + \frac{d}{d x} [- x^{2}]$

Step 1.3.2.3

Multiply $4$ by $5$ .

$20 x^{3} + \frac{d}{d x} [- x^{2}]$

Step 1.3.3

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

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Step 1.3.3.1

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

$20 x^{3} - \frac{d}{d x} [x^{2}]$

Step 1.3.3.2

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

$20 x^{3} - (2 x)$

Step 1.3.3.3

Multiply $2$ by $- 1$ .

$20 x^{3} - 2 x$

Step 1.4

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

$2 y y' = 20 x^{3} - 2 x$

Step 1.5

Divide each term in $2 y y' = 20 x^{3} - 2 x$ by $2 y$ and simplify.

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Step 1.5.1

Divide each term in $2 y y' = 20 x^{3} - 2 x$ by $2 y$ .

$\frac{2 y y'}{2 y} = \frac{20 x^{3}}{2 y} + \frac{- 2 x}{2 y}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $2$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{2 y y'}{2 y} = \frac{20 x^{3}}{2 y} + \frac{- 2 x}{2 y}$

Step 1.5.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{20 x^{3}}{2 y} + \frac{- 2 x}{2 y}$

Step 1.5.2.2

Cancel the common factor of $y$ .

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Step 1.5.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{20 x^{3}}{2 y} + \frac{- 2 x}{2 y}$

Step 1.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{20 x^{3}}{2 y} + \frac{- 2 x}{2 y}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Simplify each term.

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Step 1.5.3.1.1

Cancel the common factor of $20$ and $2$ .

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Step 1.5.3.1.1.1

Factor $2$ out of $20 x^{3}$ .

$y' = \frac{2 (10 x^{3})}{2 y} + \frac{- 2 x}{2 y}$

Step 1.5.3.1.1.2

Cancel the common factors.

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Step 1.5.3.1.1.2.1

Factor $2$ out of $2 y$ .

$y' = \frac{2 (10 x^{3})}{2 (y)} + \frac{- 2 x}{2 y}$

Step 1.5.3.1.1.2.2

Cancel the common factor.

$y' = \frac{2 (10 x^{3})}{2 y} + \frac{- 2 x}{2 y}$

Step 1.5.3.1.1.2.3

Rewrite the expression.

$y' = \frac{10 x^{3}}{y} + \frac{- 2 x}{2 y}$

Step 1.5.3.1.2

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

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Step 1.5.3.1.2.1

Factor $2$ out of $- 2 x$ .

$y' = \frac{10 x^{3}}{y} + \frac{2 (- x)}{2 y}$

Step 1.5.3.1.2.2

Cancel the common factors.

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Step 1.5.3.1.2.2.1

Factor $2$ out of $2 y$ .

$y' = \frac{10 x^{3}}{y} + \frac{2 (- x)}{2 (y)}$

Step 1.5.3.1.2.2.2

Cancel the common factor.

$y' = \frac{10 x^{3}}{y} + \frac{2 (- x)}{2 y}$

Step 1.5.3.1.2.2.3

Rewrite the expression.

$y' = \frac{10 x^{3}}{y} + \frac{- x}{y}$

Step 1.5.3.1.3

Move the negative in front of the fraction.

$y' = \frac{10 x^{3}}{y} - \frac{x}{y}$

Step 1.6

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

$\frac{d y}{d x} = \frac{10 x^{3}}{y} - \frac{x}{y}$

Step 1.7

Evaluate at $x = - 1$ and $y = 2$ .

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Step 1.7.1

Replace the variable $x$ with $- 1$ in the expression.

$\frac{10 {(- 1)}^{3}}{y} - \frac{- 1}{y}$

Step 1.7.2

Replace the variable $y$ with $2$ in the expression.

$\frac{10 {(- 1)}^{3}}{2} - \frac{- 1}{2}$

Step 1.7.3

Combine the numerators over the common denominator.

$\frac{10 {(- 1)}^{3} + 1}{2}$

Step 1.7.4

Simplify each term.

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Step 1.7.4.1

Raise $- 1$ to the power of $3$ .

$\frac{10 \cdot - 1 + 1}{2}$

Step 1.7.4.2

Multiply $10$ by $- 1$ .

$\frac{- 10 + 1}{2}$

Step 1.7.5

Simplify the expression.

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Step 1.7.5.1

Add $- 10$ and $1$ .

$\frac{- 9}{2}$

Step 1.7.5.2

Move the negative in front of the fraction.

$- \frac{9}{2}$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{9}{2} \cdot (x + 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 - \frac{9}{2} \cdot (x + 1)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y - 2 = - \frac{9}{2} x - \frac{9}{2} \cdot 1$

Step 2.3.1.2.2

Combine $x$ and $\frac{9}{2}$ .

$y - 2 = - \frac{x \cdot 9}{2} - \frac{9}{2} \cdot 1$

Step 2.3.1.2.3

Multiply $- 1$ by $1$ .

$y - 2 = - \frac{x \cdot 9}{2} - \frac{9}{2}$

Step 2.3.1.3

Move $9$ to the left of $x$ .

$y - 2 = - \frac{9 x}{2} - \frac{9}{2}$

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 $2$ to both sides of the equation.

$y = - \frac{9 x}{2} - \frac{9}{2} + 2$

Step 2.3.2.2

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

$y = - \frac{9 x}{2} - \frac{9}{2} + 2 \cdot \frac{2}{2}$

Step 2.3.2.3

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

$y = - \frac{9 x}{2} - \frac{9}{2} + \frac{2 \cdot 2}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = - \frac{9 x}{2} + \frac{- 9 + 2 \cdot 2}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $2$ by $2$ .

$y = - \frac{9 x}{2} + \frac{- 9 + 4}{2}$

Step 2.3.2.5.2

Add $- 9$ and $4$ .

$y = - \frac{9 x}{2} + \frac{- 5}{2}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = - \frac{9 x}{2} - \frac{5}{2}$

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{9}{2} x) - \frac{5}{2}$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{9}{2} x - \frac{5}{2}$

Step 3