Calculus Examples

$y^{2} = x^{3} - 4 x + 1$ at $(- 2, 1)$

Step 1

Find the first derivative and evaluate at $x = - 2$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{2}) = \frac{d}{d x} (x^{3} - 4 x + 1)$

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 $x^{3} - 4 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

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 $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [1]$

Step 1.3.2

Evaluate $\frac{d}{d x} [- 4 x]$ .

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Step 1.3.2.1

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

$3 x^{2} - 4 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

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 $n = 1$ .

$3 x^{2} - 4 \cdot 1 + \frac{d}{d x} [1]$

Step 1.3.2.3

Multiply $- 4$ by $1$ .

$3 x^{2} - 4 + \frac{d}{d x} [1]$

Step 1.3.3

Differentiate using the Constant Rule.

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Step 1.3.3.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 x^{2} - 4 + 0$

Step 1.3.3.2

Add $3 x^{2} - 4$ and $0$ .

$3 x^{2} - 4$

Step 1.4

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

$2 y y' = 3 x^{2} - 4$

Step 1.5

Divide each term in $2 y y' = 3 x^{2} - 4$ by $2 y$ and simplify.

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Step 1.5.1

Divide each term in $2 y y' = 3 x^{2} - 4$ by $2 y$ .

$\frac{2 y y'}{2 y} = \frac{3 x^{2}}{2 y} + \frac{- 4}{2 y}$

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.

$\frac{2 y y'}{2 y} = \frac{3 x^{2}}{2 y} + \frac{- 4}{2 y}$

Step 1.5.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{3 x^{2}}{2 y} + \frac{- 4}{2 y}$

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.

$\frac{y y'}{y} = \frac{3 x^{2}}{2 y} + \frac{- 4}{2 y}$

Step 1.5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{3 x^{2}}{2 y} + \frac{- 4}{2 y}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Simplify each term.

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Step 1.5.3.1.1

Cancel the common factor of $- 4$ and $2$ .

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Step 1.5.3.1.1.1

Factor $2$ out of $- 4$ .

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

Step 1.5.3.1.1.2

Cancel the common factors.

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Step 1.5.3.1.1.2.1

Factor $2$ out of $2 y$ .

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

Step 1.5.3.1.1.2.2

Cancel the common factor.

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

Step 1.5.3.1.1.2.3

Rewrite the expression.

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

Step 1.5.3.1.2

Move the negative in front of the fraction.

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

Step 1.6

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

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

Step 1.7

Evaluate at $x = - 2$ and $y = 1$ .

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.7.3

Simplify each term.

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Step 1.7.3.1

Cancel the common factor of ${(- 2)}^{2}$ and $2$ .

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Step 1.7.3.1.1

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 1.7.3.1.2

Apply the product rule to $- 1 (2)$ .

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

Step 1.7.3.1.3

Raise $- 1$ to the power of $2$ .

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

Step 1.7.3.1.4

Multiply $2^{2}$ by $1$ .

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

Step 1.7.3.1.5

Factor $2$ out of $3 \cdot 2^{2}$ .

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

Step 1.7.3.1.6

Cancel the common factors.

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Step 1.7.3.1.6.1

Cancel the common factor.

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

Step 1.7.3.1.6.2

Rewrite the expression.

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

Step 1.7.3.1.6.3

Divide $3 \cdot 2$ by $1$ .

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

Step 1.7.3.2

Multiply $3$ by $2$ .

$6 - \frac{2}{1}$

Step 1.7.3.3

Divide $2$ by $1$ .

$6 - 1 \cdot 2$

Step 1.7.3.4

Multiply $- 1$ by $2$ .

$6 - 2$

Step 1.7.4

Subtract $2$ from $6$ .

$4$

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

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

Step 2.2

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

$y - 1 = 4 \cdot (x + 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $4 \cdot (x + 2)$ .

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = 4 \cdot (x + 2)$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = 4 x + 4 \cdot 2$

Step 2.3.1.4

Multiply $4$ by $2$ .

$y - 1 = 4 x + 8$

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 $1$ to both sides of the equation.

$y = 4 x + 8 + 1$

Step 2.3.2.2

Add $8$ and $1$ .

$y = 4 x + 9$

Step 3