Calculus Examples

$y^{2} = x^{3} + 3 x^{2}$ , $(1, 2)$

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.

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

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

Differentiate.

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Step 1.3.1.1

By the Sum Rule, the derivative of $x^{3} + 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}]$ .

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

Step 1.3.1.2

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

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

Step 1.3.2

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

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Step 1.3.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

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

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 $n = 2$ .

$3 x^{2} + 3 (2 x)$

Step 1.3.2.3

Multiply $2$ by $3$ .

$3 x^{2} + 6 x$

Step 1.4

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

$2 y y' = 3 x^{2} + 6 x$

Step 1.5

Divide each term in $2 y y' = 3 x^{2} + 6 x$ by $2 y$ and simplify.

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Step 1.5.1

Divide each term in $2 y y' = 3 x^{2} + 6 x$ by $2 y$ .

$\frac{2 y y'}{2 y} = \frac{3 x^{2}}{2 y} + \frac{6 x}{2 y}$

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.

$\frac{2 y y'}{2 y} = \frac{3 x^{2}}{2 y} + \frac{6 x}{2 y}$

Step 1.5.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{3 x^{2}}{2 y} + \frac{6 x}{2 y}$

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.

$\frac{y y'}{y} = \frac{3 x^{2}}{2 y} + \frac{6 x}{2 y}$

Step 1.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{3 x^{2}}{2 y} + \frac{6 x}{2 y}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Cancel the common factor of $6$ and $2$ .

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Step 1.5.3.1.1

Factor $2$ out of $6 x$ .

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

Step 1.5.3.1.2

Cancel the common factors.

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Step 1.5.3.1.2.1

Factor $2$ out of $2 y$ .

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

Step 1.5.3.1.2.2

Cancel the common factor.

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

Step 1.5.3.1.2.3

Rewrite the expression.

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

Step 1.6

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

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

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.

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

Step 1.7.2

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

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

Step 1.7.3

Multiply $3$ by $1$ .

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

Step 1.7.4

Simplify each term.

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Step 1.7.4.1

One to any power is one.

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

Step 1.7.4.2

Multiply $2$ by $2$ .

$\frac{3 \cdot 1}{4} + \frac{3}{2}$

Step 1.7.4.3

Multiply $3$ by $1$ .

$\frac{3}{4} + \frac{3}{2}$

Step 1.7.5

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{4} + \frac{3}{2} \cdot \frac{2}{2}$

Step 1.7.6

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.7.6.1

Multiply $\frac{3}{2}$ by $\frac{2}{2}$ .

$\frac{3}{4} + \frac{3 \cdot 2}{2 \cdot 2}$

Step 1.7.6.2

Multiply $2$ by $2$ .

$\frac{3}{4} + \frac{3 \cdot 2}{4}$

Step 1.7.7

Combine the numerators over the common denominator.

$\frac{3 + 3 \cdot 2}{4}$

Step 1.7.8

Simplify the numerator.

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Step 1.7.8.1

Multiply $3$ by $2$ .

$\frac{3 + 6}{4}$

Step 1.7.8.2

Add $3$ and $6$ .

$\frac{9}{4}$

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 $\frac{9}{4}$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = \frac{9}{4} \cdot (x - (1))$

Step 2.2

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

$y - 2 = \frac{9}{4} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $\frac{9}{4} \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 2 = 0 + 0 + \frac{9}{4} \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 2 = \frac{9}{4} \cdot (x - 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 2 = \frac{9}{4} x + \frac{9}{4} \cdot - 1$

Step 2.3.1.4

Combine $\frac{9}{4}$ and $x$ .

$y - 2 = \frac{9 x}{4} + \frac{9}{4} \cdot - 1$

Step 2.3.1.5

Multiply $\frac{9}{4} \cdot - 1$ .

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Step 2.3.1.5.1

Combine $\frac{9}{4}$ and $- 1$ .

$y - 2 = \frac{9 x}{4} + \frac{9 \cdot - 1}{4}$

Step 2.3.1.5.2

Multiply $9$ by $- 1$ .

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

Step 2.3.1.6

Move the negative in front of the fraction.

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

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.

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

Step 2.3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 2.3.2.3

Combine $2$ and $\frac{4}{4}$ .

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

Step 2.3.2.4

Combine the numerators over the common denominator.

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

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $2$ by $4$ .

$y = \frac{9 x}{4} + \frac{- 9 + 8}{4}$

Step 2.3.2.5.2

Add $- 9$ and $8$ .

$y = \frac{9 x}{4} + \frac{- 1}{4}$

Step 2.3.2.6

Move the negative in front of the fraction.

$y = \frac{9 x}{4} - \frac{1}{4}$

Step 2.3.3

Reorder terms.

$y = \frac{9}{4} x - \frac{1}{4}$

Step 3