Calculus Examples

$x^{2} = y^{3}$ ; $(64, 16)$

Step 1

Find the first derivative and evaluate at $x = 64$ and $y = 16$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

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

Step 1.2

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

$2 x$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 1.3.1.1

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

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [y]$

Step 1.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 3$ .

$3 u^{2} \frac{d}{d x} [y]$

Step 1.3.1.3

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

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

Step 1.3.2

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

$3 y^{2} y'$

Step 1.4

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

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

Step 1.5

Solve for $y'$ .

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Step 1.5.1

Rewrite the equation as $3 y^{2} y' = 2 x$ .

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

Simplify the left side.

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Step 1.5.2.2.1

Cancel the common factor of $3$ .

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Step 1.5.2.2.1.1

Cancel the common factor.

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

Step 1.5.2.2.1.2

Rewrite the expression.

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

Step 1.5.2.2.2

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

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Step 1.5.2.2.2.1

Cancel the common factor.

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

Step 1.5.2.2.2.2

Divide $y'$ by $1$ .

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

Step 1.6

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

$\frac{d y}{d x} = \frac{2 x}{3 y^{2}}$

Step 1.7

Evaluate at $x = 64$ and $y = 16$ .

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Step 1.7.1

Replace the variable $x$ with $64$ in the expression.

$\frac{2 (64)}{3 y^{2}}$

Step 1.7.2

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

$\frac{2 (64)}{3 {(16)}^{2}}$

Step 1.7.3

Simplify the expression.

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Step 1.7.3.1

Multiply $2$ by $64$ .

$\frac{128}{3 \cdot 16^{2}}$

Step 1.7.3.2

Raise $16$ to the power of $2$ .

$\frac{128}{3 \cdot 256}$

Step 1.7.3.3

Multiply $3$ by $256$ .

$\frac{128}{768}$

Step 1.7.4

Cancel the common factor of $128$ and $768$ .

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Step 1.7.4.1

Factor $128$ out of $128$ .

$\frac{128 (1)}{768}$

Step 1.7.4.2

Cancel the common factors.

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Step 1.7.4.2.1

Factor $128$ out of $768$ .

$\frac{128 \cdot 1}{128 \cdot 6}$

Step 1.7.4.2.2

Cancel the common factor.

$\frac{128 \cdot 1}{128 \cdot 6}$

Step 1.7.4.2.3

Rewrite the expression.

$\frac{1}{6}$

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 $\frac{1}{6}$ and a given point $(64, 16)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (16) = \frac{1}{6} \cdot (x - (64))$

Step 2.2

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

$y - 16 = \frac{1}{6} \cdot (x - 64)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{1}{6} \cdot (x - 64)$ .

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Step 2.3.1.1

Rewrite.

$y - 16 = 0 + 0 + \frac{1}{6} \cdot (x - 64)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 16 = \frac{1}{6} \cdot (x - 64)$

Step 2.3.1.3

Apply the distributive property.

$y - 16 = \frac{1}{6} x + \frac{1}{6} \cdot - 64$

Step 2.3.1.4

Combine $\frac{1}{6}$ and $x$ .

$y - 16 = \frac{x}{6} + \frac{1}{6} \cdot - 64$

Step 2.3.1.5

Cancel the common factor of $2$ .

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Step 2.3.1.5.1

Factor $2$ out of $6$ .

$y - 16 = \frac{x}{6} + \frac{1}{2 (3)} \cdot - 64$

Step 2.3.1.5.2

Factor $2$ out of $- 64$ .

$y - 16 = \frac{x}{6} + \frac{1}{2 \cdot 3} \cdot (2 \cdot - 32)$

Step 2.3.1.5.3

Cancel the common factor.

$y - 16 = \frac{x}{6} + \frac{1}{2 \cdot 3} \cdot (2 \cdot - 32)$

Step 2.3.1.5.4

Rewrite the expression.

$y - 16 = \frac{x}{6} + \frac{1}{3} \cdot - 32$

Step 2.3.1.6

Combine $\frac{1}{3}$ and $- 32$ .

$y - 16 = \frac{x}{6} + \frac{- 32}{3}$

Step 2.3.1.7

Move the negative in front of the fraction.

$y - 16 = \frac{x}{6} - \frac{32}{3}$

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 $16$ to both sides of the equation.

$y = \frac{x}{6} - \frac{32}{3} + 16$

Step 2.3.2.2

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{x}{6} - \frac{32}{3} + 16 \cdot \frac{3}{3}$

Step 2.3.2.3

Combine $16$ and $\frac{3}{3}$ .

$y = \frac{x}{6} - \frac{32}{3} + \frac{16 \cdot 3}{3}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = \frac{x}{6} + \frac{- 32 + 16 \cdot 3}{3}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $16$ by $3$ .

$y = \frac{x}{6} + \frac{- 32 + 48}{3}$

Step 2.3.2.5.2

Add $- 32$ and $48$ .

$y = \frac{x}{6} + \frac{16}{3}$

Step 2.3.3

Reorder terms.

$y = \frac{1}{6} x + \frac{16}{3}$

Step 3