Calculus Examples

$f (x) = 2 x^{2} - 4 x + 5$ at the point $(- 1, 11)$

Step 1

Find the first derivative and evaluate at $x = - 1$ and $y = 11$ to find the slope of the tangent line.

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Step 1.1

By the Sum Rule, the derivative of $2 x^{2} - 4 x + 5$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]$

Step 1.2

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

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Step 1.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]$

Step 1.2.2

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

$2 (2 x) + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]$

Step 1.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [5]$

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

$4 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.3.2

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

$4 x - 4 \cdot 1 + \frac{d}{d x} [5]$

Step 1.3.3

Multiply $- 4$ by $1$ .

$4 x - 4 + \frac{d}{d x} [5]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$4 x - 4 + 0$

Step 1.4.2

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

$4 x - 4$

Step 1.5

Evaluate the derivative at $x = - 1$ .

$4 (- 1) - 4$

Step 1.6

Simplify.

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Step 1.6.1

Multiply $4$ by $- 1$ .

$- 4 - 4$

Step 1.6.2

Subtract $4$ from $- 4$ .

$- 8$

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 $- 8$ and a given point $(- 1, 11)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (11) = - 8 \cdot (x - (- 1))$

Step 2.2

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

$y - 11 = - 8 \cdot (x + 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 8 \cdot (x + 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 11 = 0 + 0 - 8 \cdot (x + 1)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 11 = - 8 \cdot (x + 1)$

Step 2.3.1.3

Apply the distributive property.

$y - 11 = - 8 x - 8 \cdot 1$

Step 2.3.1.4

Multiply $- 8$ by $1$ .

$y - 11 = - 8 x - 8$

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 $11$ to both sides of the equation.

$y = - 8 x - 8 + 11$

Step 2.3.2.2

Add $- 8$ and $11$ .

$y = - 8 x + 3$

Step 3