Calculus Examples

$y^{2} - x^{2} = 16$ ; $(2, 2 \sqrt{5})$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = 2 \sqrt{5}$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2} - x^{2}) = \frac{d}{d x} (16)$

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

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

Step 1.2.2

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

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Step 1.2.2.1

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 1.2.2.1.1

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

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

Step 1.2.2.1.2

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

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

Step 1.2.2.1.3

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

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

Step 1.2.2.2

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

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

$2 y y' - (2 x)$

Step 1.2.3.3

Multiply $2$ by $- 1$ .

$2 y y' - 2 x$

Step 1.3

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$0$

Step 1.4

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

$2 y y' - 2 x = 0$

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Add $2 x$ to both sides of the equation.

$2 y y' = 2 x$

Step 1.5.2

Divide each term in $2 y y' = 2 x$ by $2 y$ and simplify.

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Step 1.5.2.1

Divide each term in $2 y y' = 2 x$ by $2 y$ .

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

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $2$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.1.2

Rewrite the expression.

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

Step 1.5.2.2.2

Cancel the common factor of $y$ .

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Step 1.5.2.2.2.1

Cancel the common factor.

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

Step 1.5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 1.5.2.3

Simplify the right side.

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Step 1.5.2.3.1

Cancel the common factor of $2$ .

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Step 1.5.2.3.1.1

Cancel the common factor.

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

Step 1.5.2.3.1.2

Rewrite the expression.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = 2$ and $y = 2 \sqrt{5}$ .

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Step 1.7.1

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

$\frac{2}{y}$

Step 1.7.2

Replace the variable $y$ with $2 \sqrt{5}$ in the expression.

$\frac{2}{2 \sqrt{5}}$

Step 1.7.3

Cancel the common factor of $2$ .

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Step 1.7.3.1

Cancel the common factor.

$\frac{2}{2 \sqrt{5}}$

Step 1.7.3.2

Rewrite the expression.

$\frac{1}{\sqrt{5}}$

Step 1.7.4

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 1.7.5

Combine and simplify the denominator.

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Step 1.7.5.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 1.7.5.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 1.7.5.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 1.7.5.4

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

$\frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 1.7.5.5

Add $1$ and $1$ .

$\frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 1.7.5.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 1.7.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 1.7.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 1.7.5.6.3

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

$\frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.7.5.6.4

Cancel the common factor of $2$ .

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Step 1.7.5.6.4.1

Cancel the common factor.

$\frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 1.7.5.6.4.2

Rewrite the expression.

$\frac{\sqrt{5}}{5^{1}}$

Step 1.7.5.6.5

Evaluate the exponent.

$\frac{\sqrt{5}}{5}$

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 $\frac{\sqrt{5}}{5}$ and a given point $(2, 2 \sqrt{5})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{\sqrt{5}}{5} \cdot (x - 2)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 \sqrt{5} = 0 + 0 + \frac{\sqrt{5}}{5} \cdot (x - 2)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

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

Step 2.3.1.2.2

Combine $\frac{\sqrt{5}}{5}$ and $x$ .

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

Step 2.3.1.2.3

Combine $\frac{\sqrt{5}}{5}$ and $- 2$ .

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

Step 2.3.1.3

Simplify each term.

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Step 2.3.1.3.1

Move $- 2$ to the left of $\sqrt{5}$ .

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

Step 2.3.1.3.2

Move the negative in front of the fraction.

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

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 $2 \sqrt{5}$ to both sides of the equation.

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

Step 2.3.2.2

To write $2 \sqrt{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

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

Step 2.3.2.3

Combine $2 \sqrt{5}$ and $\frac{5}{5}$ .

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

Step 2.3.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.5

Combine the numerators over the common denominator.

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

Step 2.3.2.6

Multiply $5$ by $2$ .

$y = \frac{\sqrt{5} x - 2 \sqrt{5} + 10 \sqrt{5}}{5}$

Step 2.3.2.7

Add $- 2 \sqrt{5}$ and $10 \sqrt{5}$ .

$y = \frac{\sqrt{5} x + 8 \sqrt{5}}{5}$

Step 2.3.2.8

Split the fraction $\frac{\sqrt{5} x + 8 \sqrt{5}}{5}$ into two fractions.

$y = \frac{\sqrt{5} x}{5} + \frac{8 \sqrt{5}}{5}$

Step 2.3.3

Reorder terms.

$y = \frac{\sqrt{5}}{5} x + \frac{8 \sqrt{5}}{5}$

Step 3