Calculus Examples

$\sqrt{x} + \sqrt{y} = 3$ , $(4, 1)$

Step 1

Find the first derivative and evaluate at $x = 4$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Rewrite the left side with rational exponents.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{\frac{1}{2}} + y^{\frac{1}{2}}) = \frac{d}{d x} (3)$

Step 1.3

Differentiate the left side of the equation.

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Step 1.3.1

By the Sum Rule, the derivative of $x^{\frac{1}{2}} + y^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [y^{\frac{1}{2}}]$ .

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

Step 1.3.2

Evaluate $\frac{d}{d x} [x^{\frac{1}{2}}]$ .

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Step 1.3.2.1

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

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

Step 1.3.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.3.2.3

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

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

Step 1.3.2.4

Combine the numerators over the common denominator.

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

Step 1.3.2.5

Simplify the numerator.

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Step 1.3.2.5.1

Multiply $- 1$ by $2$ .

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

Step 1.3.2.5.2

Subtract $2$ from $1$ .

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

Step 1.3.2.6

Move the negative in front of the fraction.

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

Step 1.3.3

Evaluate $\frac{d}{d x} [y^{\frac{1}{2}}]$ .

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Step 1.3.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = y$ .

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Step 1.3.3.1.1

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

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

Step 1.3.3.1.2

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

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

Step 1.3.3.1.3

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

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

Step 1.3.3.2

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

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{2} y^{\frac{1}{2} - 1} y'$

Step 1.3.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}} y'$

Step 1.3.3.4

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

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{2} y^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} y'$

Step 1.3.3.5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}} y'$

Step 1.3.3.6

Simplify the numerator.

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Step 1.3.3.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{2} y^{\frac{1 - 2}{2}} y'$

Step 1.3.3.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{2} y^{\frac{- 1}{2}} y'$

Step 1.3.3.7

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{1}{2} y^{- \frac{1}{2}} y'$

Step 1.3.3.8

Combine $\frac{1}{2}$ and $y^{- \frac{1}{2}}$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{y^{- \frac{1}{2}}}{2} y'$

Step 1.3.3.9

Combine $\frac{y^{- \frac{1}{2}}}{2}$ and $y'$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{y^{- \frac{1}{2}} y'}{2}$

Step 1.3.3.10

Move $y^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} x^{- \frac{1}{2}} + \frac{y'}{2 y^{\frac{1}{2}}}$

Step 1.3.4

Simplify.

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Step 1.3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} + \frac{y'}{2 y^{\frac{1}{2}}}$

Step 1.3.4.2

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$ .

$\frac{1}{2 x^{\frac{1}{2}}} + \frac{y'}{2 y^{\frac{1}{2}}}$

Step 1.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0$

Step 1.5

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

$\frac{1}{2 x^{\frac{1}{2}}} + \frac{y'}{2 y^{\frac{1}{2}}} = 0$

Step 1.6

Solve for $y'$ .

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Step 1.6.1

Subtract $\frac{1}{2 x^{\frac{1}{2}}}$ from both sides of the equation.

$\frac{y'}{2 y^{\frac{1}{2}}} = - \frac{1}{2 x^{\frac{1}{2}}}$

Step 1.6.2

Multiply both sides by $2 y^{\frac{1}{2}}$ .

$\frac{y'}{2 y^{\frac{1}{2}}} (2 y^{\frac{1}{2}}) = - \frac{1}{2 x^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$

Step 1.6.3

Simplify.

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Step 1.6.3.1

Simplify the left side.

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Step 1.6.3.1.1

Simplify $\frac{y'}{2 y^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$ .

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Step 1.6.3.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{y'}{2 y^{\frac{1}{2}}} y^{\frac{1}{2}} = - \frac{1}{2 x^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$

Step 1.6.3.1.1.2

Cancel the common factor of $2$ .

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Step 1.6.3.1.1.2.1

Cancel the common factor.

$2 \frac{y'}{2 y^{\frac{1}{2}}} y^{\frac{1}{2}} = - \frac{1}{2 x^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$

Step 1.6.3.1.1.2.2

Rewrite the expression.

$\frac{y'}{y^{\frac{1}{2}}} y^{\frac{1}{2}} = - \frac{1}{2 x^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$

Step 1.6.3.1.1.3

Cancel the common factor of $y^{\frac{1}{2}}$ .

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Step 1.6.3.1.1.3.1

Cancel the common factor.

$\frac{y'}{y^{\frac{1}{2}}} y^{\frac{1}{2}} = - \frac{1}{2 x^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$

Step 1.6.3.1.1.3.2

Rewrite the expression.

$y' = - \frac{1}{2 x^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$

Step 1.6.3.2

Simplify the right side.

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Step 1.6.3.2.1

Simplify $- \frac{1}{2 x^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$ .

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Step 1.6.3.2.1.1

Cancel the common factor of $2$ .

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Step 1.6.3.2.1.1.1

Move the leading negative in $- \frac{1}{2 x^{\frac{1}{2}}}$ into the numerator.

$y' = \frac{- 1}{2 x^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$

Step 1.6.3.2.1.1.2

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$y' = \frac{- 1}{2 (x^{\frac{1}{2}})} (2 y^{\frac{1}{2}})$

Step 1.6.3.2.1.1.3

Factor $2$ out of $2 y^{\frac{1}{2}}$ .

$y' = \frac{- 1}{2 (x^{\frac{1}{2}})} (2 (y^{\frac{1}{2}}))$

Step 1.6.3.2.1.1.4

Cancel the common factor.

$y' = \frac{- 1}{2 x^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$

Step 1.6.3.2.1.1.5

Rewrite the expression.

$y' = \frac{- 1}{x^{\frac{1}{2}}} y^{\frac{1}{2}}$

Step 1.6.3.2.1.2

Combine $\frac{- 1}{x^{\frac{1}{2}}}$ and $y^{\frac{1}{2}}$ .

$y' = \frac{- y^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 1.6.3.2.1.3

Move the negative in front of the fraction.

$y' = - \frac{y^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 1.7

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

$\frac{d y}{d x} = - \frac{y^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 1.8

Evaluate at $x = 4$ and $y = 1$ .

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Step 1.8.1

Replace the variable $x$ with $4$ in the expression.

$- \frac{y^{\frac{1}{2}}}{{(4)}^{\frac{1}{2}}}$

Step 1.8.2

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

$- \frac{{(1)}^{\frac{1}{2}}}{{(4)}^{\frac{1}{2}}}$

Step 1.8.3

One to any power is one.

$- \frac{1}{4^{\frac{1}{2}}}$

Step 1.8.4

Simplify the denominator.

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Step 1.8.4.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.8.4.2

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

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

Step 1.8.4.3

Cancel the common factor of $2$ .

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Step 1.8.4.3.1

Cancel the common factor.

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

Step 1.8.4.3.2

Rewrite the expression.

$- \frac{1}{2^{1}}$

Step 1.8.4.4

Evaluate the exponent.

$- \frac{1}{2}$

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 $- \frac{1}{2}$ and a given point $(4, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{1}{2} \cdot (x - (4))$

Step 2.2

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

$y - 1 = - \frac{1}{2} \cdot (x - 4)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{1}{2} \cdot (x - 4)$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{1}{2} \cdot (x - 4)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = - \frac{1}{2} \cdot (x - 4)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = - \frac{1}{2} x - \frac{1}{2} \cdot - 4$

Step 2.3.1.4

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

$y - 1 = - \frac{x}{2} - \frac{1}{2} \cdot - 4$

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y - 1 = - \frac{x}{2} + \frac{- 1}{2} \cdot - 4$

Step 2.3.1.5.2

Factor $2$ out of $- 4$ .

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

Step 2.3.1.5.3

Cancel the common factor.

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

Step 2.3.1.5.4

Rewrite the expression.

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

Step 2.3.1.6

Multiply $- 1$ by $- 2$ .

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

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 $1$ to both sides of the equation.

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

Step 2.3.2.2

Add $2$ and $1$ .

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

Step 2.3.3

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

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Step 2.3.3.1

Reorder terms.

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

Step 2.3.3.2

Remove parentheses.

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

Step 3