Calculus Examples

$f (x) = - \frac{3}{2} x^{2} + 4 x + 4$ ; $(- 1, - \frac{3}{2})$

Step 1

Find the first derivative and evaluate at $x = - 1$ and $y = - \frac{3}{2}$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of $- \frac{3}{2} x^{2} + 4 x + 4$ with respect to $x$ is $\frac{d}{d x} [- \frac{3}{2} x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

$- \frac{3}{2} (2 x) + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$

Step 1.2.3

Multiply $2$ by $- 1$ .

$- 2 (\frac{3}{2}) x + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$

Step 1.2.4

Combine $- 2$ and $\frac{3}{2}$ .

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

Step 1.2.5

Multiply $- 2$ by $3$ .

$\frac{- 6}{2} x + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$

Step 1.2.6

Combine $\frac{- 6}{2}$ and $x$ .

$\frac{- 6 x}{2} + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$

Step 1.2.7

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

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Step 1.2.7.1

Factor $2$ out of $- 6 x$ .

$\frac{2 (- 3 x)}{2} + \frac{d}{d x} [4 x] + \frac{d}{d x} [4]$

Step 1.2.7.2

Cancel the common factors.

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Step 1.2.7.2.1

Factor $2$ out of $2$ .

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

Step 1.2.7.2.2

Cancel the common factor.

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

Step 1.2.7.2.3

Rewrite the expression.

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

Step 1.2.7.2.4

Divide $- 3 x$ by $1$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $4$ by $1$ .

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

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- 3 x + 4 + 0$

Step 1.4.2

Add $- 3 x + 4$ and $0$ .

$- 3 x + 4$

Step 1.5

Evaluate the derivative at $x = - 1$ .

$- 3 \cdot - 1 + 4$

Step 1.6

Simplify.

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Step 1.6.1

Multiply $- 3$ by $- 1$ .

$3 + 4$

Step 1.6.2

Add $3$ and $4$ .

$7$

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 $7$ and a given point $(- 1, - \frac{3}{2})$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- \frac{3}{2}) = 7 \cdot (x - (- 1))$

Step 2.2

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

$y + \frac{3}{2} = 7 \cdot (x + 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $7 \cdot (x + 1)$ .

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Step 2.3.1.1

Rewrite.

$y + \frac{3}{2} = 0 + 0 + 7 \cdot (x + 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y + \frac{3}{2} = 7 \cdot (x + 1)$

Step 2.3.1.3

Apply the distributive property.

$y + \frac{3}{2} = 7 x + 7 \cdot 1$

Step 2.3.1.4

Multiply $7$ by $1$ .

$y + \frac{3}{2} = 7 x + 7$

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

Subtract $\frac{3}{2}$ from both sides of the equation.

$y = 7 x + 7 - \frac{3}{2}$

Step 2.3.2.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 7 x + 7 \cdot \frac{2}{2} - \frac{3}{2}$

Step 2.3.2.3

Combine $7$ and $\frac{2}{2}$ .

$y = 7 x + \frac{7 \cdot 2}{2} - \frac{3}{2}$

Step 2.3.2.4

Combine the numerators over the common denominator.

$y = 7 x + \frac{7 \cdot 2 - 3}{2}$

Step 2.3.2.5

Simplify the numerator.

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Step 2.3.2.5.1

Multiply $7$ by $2$ .

$y = 7 x + \frac{14 - 3}{2}$

Step 2.3.2.5.2

Subtract $3$ from $14$ .

$y = 7 x + \frac{11}{2}$

Step 3