Calculus Examples

$4 y^{2} - \sqrt{x} = 12$ , $(16, 2)$

Step 1

Find the first derivative and evaluate at $x = 16$ and $y = 2$ to find the slope of the tangent line.

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Step 1.1

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

$4 y^{2} - x^{\frac{1}{2}} = 12$

Step 1.2

Differentiate both sides of the equation.

$\frac{d}{d x} (4 y^{2} - x^{\frac{1}{2}}) = \frac{d}{d x} (12)$

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 $4 y^{2} - x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- x^{\frac{1}{2}}]$ .

$\frac{d}{d x} [4 y^{2}] + \frac{d}{d x} [- x^{\frac{1}{2}}]$

Step 1.3.2

Evaluate $\frac{d}{d x} [4 y^{2}]$ .

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Step 1.3.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 y^{2}$ with respect to $x$ is $4 \frac{d}{d x} [y^{2}]$ .

$4 \frac{d}{d x} [y^{2}] + \frac{d}{d x} [- x^{\frac{1}{2}}]$

Step 1.3.2.2

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

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Step 1.3.2.2.1

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

$4 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [- x^{\frac{1}{2}}]$

Step 1.3.2.2.2

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

$4 (2 u \frac{d}{d x} [y]) + \frac{d}{d x} [- x^{\frac{1}{2}}]$

Step 1.3.2.2.3

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

$4 (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [- x^{\frac{1}{2}}]$

Step 1.3.2.3

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

$4 (2 y y') + \frac{d}{d x} [- x^{\frac{1}{2}}]$

Step 1.3.2.4

Multiply $2$ by $4$ .

$8 y y' + \frac{d}{d x} [- x^{\frac{1}{2}}]$

Step 1.3.3

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

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Step 1.3.3.1

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

$8 y y' - \frac{d}{d x} [x^{\frac{1}{2}}]$

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 $n = \frac{1}{2}$ .

$8 y y' - (\frac{1}{2} x^{\frac{1}{2} - 1})$

Step 1.3.3.3

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

$8 y y' - (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 1.3.3.4

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

$8 y y' - (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 1.3.3.5

Combine the numerators over the common denominator.

$8 y y' - (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})$

Step 1.3.3.6

Simplify the numerator.

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Step 1.3.3.6.1

Multiply $- 1$ by $2$ .

$8 y y' - (\frac{1}{2} x^{\frac{1 - 2}{2}})$

Step 1.3.3.6.2

Subtract $2$ from $1$ .

$8 y y' - (\frac{1}{2} x^{\frac{- 1}{2}})$

Step 1.3.3.7

Move the negative in front of the fraction.

$8 y y' - (\frac{1}{2} x^{- \frac{1}{2}})$

Step 1.3.3.8

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

$8 y y' - \frac{x^{- \frac{1}{2}}}{2}$

Step 1.3.3.9

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

$8 y y' - \frac{1}{2 x^{\frac{1}{2}}}$

Step 1.4

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$0$

Step 1.5

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

$8 y y' - \frac{1}{2 x^{\frac{1}{2}}} = 0$

Step 1.6

Solve for $y'$ .

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Step 1.6.1

Add $\frac{1}{2 x^{\frac{1}{2}}}$ to both sides of the equation.

$8 y y' = \frac{1}{2 x^{\frac{1}{2}}}$

Step 1.6.2

Divide each term in $8 y y' = \frac{1}{2 x^{\frac{1}{2}}}$ by $8 y$ and simplify.

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Step 1.6.2.1

Divide each term in $8 y y' = \frac{1}{2 x^{\frac{1}{2}}}$ by $8 y$ .

$\frac{8 y y'}{8 y} = \frac{\frac{1}{2 x^{\frac{1}{2}}}}{8 y}$

Step 1.6.2.2

Simplify the left side.

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Step 1.6.2.2.1

Cancel the common factor of $8$ .

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Step 1.6.2.2.1.1

Cancel the common factor.

$\frac{8 y y'}{8 y} = \frac{\frac{1}{2 x^{\frac{1}{2}}}}{8 y}$

Step 1.6.2.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{\frac{1}{2 x^{\frac{1}{2}}}}{8 y}$

Step 1.6.2.2.2

Cancel the common factor of $y$ .

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Step 1.6.2.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{\frac{1}{2 x^{\frac{1}{2}}}}{8 y}$

Step 1.6.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{1}{2 x^{\frac{1}{2}}}}{8 y}$

Step 1.6.2.3

Simplify the right side.

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Step 1.6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{1}{2 x^{\frac{1}{2}}} \cdot \frac{1}{8 y}$

Step 1.6.2.3.2

Combine.

$y' = \frac{1 \cdot 1}{2 x^{\frac{1}{2}} (8 y)}$

Step 1.6.2.3.3

Simplify the expression.

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Step 1.6.2.3.3.1

Multiply $1$ by $1$ .

$y' = \frac{1}{2 x^{\frac{1}{2}} \cdot 8 y}$

Step 1.6.2.3.3.2

Multiply $8$ by $2$ .

$y' = \frac{1}{16 x^{\frac{1}{2}} y}$

Step 1.6.2.3.3.3

Reorder factors in $\frac{1}{16 x^{\frac{1}{2}} y}$ .

$y' = \frac{1}{16 y x^{\frac{1}{2}}}$

Step 1.7

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

$\frac{d y}{d x} = \frac{1}{16 y x^{\frac{1}{2}}}$

Step 1.8

Evaluate at $x = 16$ and $y = 2$ .

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Step 1.8.1

Replace the variable $x$ with $16$ in the expression.

$\frac{1}{16 y {(16)}^{\frac{1}{2}}}$

Step 1.8.2

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

$\frac{1}{16 (2) {(16)}^{\frac{1}{2}}}$

Step 1.8.3

Multiply $16$ by ${(16)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.8.3.1

Move ${(16)}^{\frac{1}{2}}$ .

$\frac{1}{{(16)}^{\frac{1}{2}} \cdot 16 \cdot 2}$

Step 1.8.3.2

Multiply ${(16)}^{\frac{1}{2}}$ by $16$ .

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Step 1.8.3.2.1

Raise $16$ to the power of $1$ .

$\frac{1}{{(16)}^{\frac{1}{2}} \cdot 16^{1} \cdot 2}$

Step 1.8.3.2.2

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

$\frac{1}{16^{\frac{1}{2} + 1} \cdot 2}$

Step 1.8.3.3

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

$\frac{1}{16^{\frac{1}{2} + \frac{2}{2}} \cdot 2}$

Step 1.8.3.4

Combine the numerators over the common denominator.

$\frac{1}{16^{\frac{1 + 2}{2}} \cdot 2}$

Step 1.8.3.5

Add $1$ and $2$ .

$\frac{1}{16^{\frac{3}{2}} \cdot 2}$

Step 1.8.4

Simplify the denominator.

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Step 1.8.4.1

Rewrite $16$ as $2^{4}$ .

$\frac{1}{{(2^{4})}^{\frac{3}{2}} \cdot 2}$

Step 1.8.4.2

Multiply the exponents in ${(2^{4})}^{\frac{3}{2}}$ .

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Step 1.8.4.2.1

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

$\frac{1}{2^{4 (\frac{3}{2})} \cdot 2}$

Step 1.8.4.2.2

Cancel the common factor of $2$ .

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Step 1.8.4.2.2.1

Factor $2$ out of $4$ .

$\frac{1}{2^{2 (2) \frac{3}{2}} \cdot 2}$

Step 1.8.4.2.2.2

Cancel the common factor.

$\frac{1}{2^{2 \cdot 2 \frac{3}{2}} \cdot 2}$

Step 1.8.4.2.2.3

Rewrite the expression.

$\frac{1}{2^{2 \cdot 3} \cdot 2}$

Step 1.8.4.2.3

Multiply $2$ by $3$ .

$\frac{1}{2^{6} \cdot 2}$

Step 1.8.4.3

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

$\frac{1}{2^{6 + 1}}$

Step 1.8.4.4

Add $6$ and $1$ .

$\frac{1}{2^{7}}$

Step 1.8.5

Raise $2$ to the power of $7$ .

$\frac{1}{128}$

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 $\frac{1}{128}$ and a given point $(16, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = \frac{1}{128} \cdot (x - (16))$

Step 2.2

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

$y - 2 = \frac{1}{128} \cdot (x - 16)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{128} \cdot (x - 16)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 + \frac{1}{128} \cdot (x - 16)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 2 = \frac{1}{128} \cdot (x - 16)$

Step 2.3.1.3

Apply the distributive property.

$y - 2 = \frac{1}{128} x + \frac{1}{128} \cdot - 16$

Step 2.3.1.4

Combine $\frac{1}{128}$ and $x$ .

$y - 2 = \frac{x}{128} + \frac{1}{128} \cdot - 16$

Step 2.3.1.5

Cancel the common factor of $16$ .

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Step 2.3.1.5.1

Factor $16$ out of $128$ .

$y - 2 = \frac{x}{128} + \frac{1}{16 (8)} \cdot - 16$

Step 2.3.1.5.2

Factor $16$ out of $- 16$ .

$y - 2 = \frac{x}{128} + \frac{1}{16 \cdot 8} \cdot (16 \cdot - 1)$

Step 2.3.1.5.3

Cancel the common factor.

$y - 2 = \frac{x}{128} + \frac{1}{16 \cdot 8} \cdot (16 \cdot - 1)$

Step 2.3.1.5.4

Rewrite the expression.

$y - 2 = \frac{x}{128} + \frac{1}{8} \cdot - 1$

Step 2.3.1.6

Combine $\frac{1}{8}$ and $- 1$ .

$y - 2 = \frac{x}{128} + \frac{- 1}{8}$

Step 2.3.1.7

Move the negative in front of the fraction.

$y - 2 = \frac{x}{128} - \frac{1}{8}$

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.

$y = \frac{x}{128} - \frac{1}{8} + 2$

Step 2.3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{x}{128} - \frac{1}{8} + 2 \cdot \frac{8}{8}$

Step 2.3.2.3

Combine $2$ and $\frac{8}{8}$ .

$y = \frac{x}{128} - \frac{1}{8} + \frac{2 \cdot 8}{8}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{128} + \frac{- 1 + 2 \cdot 8}{8}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $2$ by $8$ .

$y = \frac{x}{128} + \frac{- 1 + 16}{8}$

Step 2.3.2.5.2

Add $- 1$ and $16$ .

$y = \frac{x}{128} + \frac{15}{8}$

Step 2.3.3

Reorder terms.

$y = \frac{1}{128} x + \frac{15}{8}$

Step 3