Calculus Examples

$y = 2 x^{2} - 8 x + 1$ , $(5, 11)$

Step 1

Find the first derivative and evaluate at $x = 5$ and $y = 11$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

By the Sum Rule, the derivative of $2 x^{2} - 8 x + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [1]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [1]$

Step 1.2.2

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

$2 (2 x) + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [1]$

Step 1.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [1]$

Step 1.3

Evaluate $\frac{d}{d x} [- 8 x]$ .

Tap for more steps...

Step 1.3.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$4 x - 8 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 1.3.2

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

$4 x - 8 \cdot 1 + \frac{d}{d x} [1]$

Step 1.3.3

Multiply $- 8$ by $1$ .

$4 x - 8 + \frac{d}{d x} [1]$

Step 1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.4.1

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

$4 x - 8 + 0$

Step 1.4.2

Add $4 x - 8$ and $0$ .

$4 x - 8$

Step 1.5

Evaluate the derivative at $x = 5$ .

$4 (5) - 8$

Step 1.6

Simplify.

Tap for more steps...

Step 1.6.1

Multiply $4$ by $5$ .

$20 - 8$

Step 1.6.2

Subtract $8$ from $20$ .

$12$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

Tap for more steps...

Step 2.1

Use the slope $12$ and a given point $(5, 11)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (11) = 12 \cdot (x - (5))$

Step 2.2

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

$y - 11 = 12 \cdot (x - 5)$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $12 \cdot (x - 5)$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y - 11 = 0 + 0 + 12 \cdot (x - 5)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 11 = 12 \cdot (x - 5)$

Step 2.3.1.3

Apply the distributive property.

$y - 11 = 12 x + 12 \cdot - 5$

Step 2.3.1.4

Multiply $12$ by $- 5$ .

$y - 11 = 12 x - 60$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 2.3.2.1

Add $11$ to both sides of the equation.

$y = 12 x - 60 + 11$

Step 2.3.2.2

Add $- 60$ and $11$ .

$y = 12 x - 49$

Step 3