Calculus Examples

$y = x^{2} - 5 x - 2$ , $(3, - 8)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = - 8$ 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 $x^{2} - 5 x - 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2]$

Step 1.1.2

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

$2 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 2]$

Step 1.2

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 1.2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$2 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]$

Step 1.2.2

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

$2 x - 5 \cdot 1 + \frac{d}{d x} [- 2]$

Step 1.2.3

Multiply $- 5$ by $1$ .

$2 x - 5 + \frac{d}{d x} [- 2]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$2 x - 5 + 0$

Step 1.3.2

Add $2 x - 5$ and $0$ .

$2 x - 5$

Step 1.4

Evaluate the derivative at $x = 3$ .

$2 (3) - 5$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $2$ by $3$ .

$6 - 5$

Step 1.5.2

Subtract $5$ from $6$ .

$1$

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 $1$ and a given point $(3, - 8)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 8) = 1 \cdot (x - (3))$

Step 2.2

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

$y + 8 = 1 \cdot (x - 3)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $x - 3$ by $1$ .

$y + 8 = x - 3$

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

Subtract $8$ from both sides of the equation.

$y = x - 3 - 8$

Step 2.3.2.2

Subtract $8$ from $- 3$ .

$y = x - 11$

Step 3