Calculus Examples

$y = x^{2} + 3 x - 9$ , $(3, 9)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = 9$ to find the slope of the tangent line.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{2} + 3 x - 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$

Step 1.1.2

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

$2 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$

Step 1.2

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

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Step 1.2.1

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

$2 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]$

Step 1.2.2

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

$2 x + 3 \cdot 1 + \frac{d}{d x} [- 9]$

Step 1.2.3

Multiply $3$ by $1$ .

$2 x + 3 + \frac{d}{d x} [- 9]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$2 x + 3 + 0$

Step 1.3.2

Add $2 x + 3$ and $0$ .

$2 x + 3$

Step 1.4

Evaluate the derivative at $x = 3$ .

$2 (3) + 3$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $2$ by $3$ .

$6 + 3$

Step 1.5.2

Add $6$ and $3$ .

$9$

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 $9$ and a given point $(3, 9)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (9) = 9 \cdot (x - (3))$

Step 2.2

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

$y - 9 = 9 \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $9 \cdot (x - 3)$ .

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Step 2.3.1.1

Rewrite.

$y - 9 = 0 + 0 + 9 \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 9 = 9 \cdot (x - 3)$

Step 2.3.1.3

Apply the distributive property.

$y - 9 = 9 x + 9 \cdot - 3$

Step 2.3.1.4

Multiply $9$ by $- 3$ .

$y - 9 = 9 x - 27$

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 $9$ to both sides of the equation.

$y = 9 x - 27 + 9$

Step 2.3.2.2

Add $- 27$ and $9$ .

$y = 9 x - 18$

Step 3

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