Calculus Examples

$f (x) = 1 - 2 x^{2}$ , $x = - 1$

Step 1

Find the corresponding $y$ -value to $x = - 1$ .

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Step 1.1

Substitute $- 1$ in for $x$ .

$y = 1 - 2 {(- 1)}^{2}$

Step 1.2

Simplify $1 - 2 {(- 1)}^{2}$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Raise $- 1$ to the power of $2$ .

$y = 1 - 2 \cdot 1$

Step 1.2.1.2

Multiply $- 2$ by $1$ .

$y = 1 - 2$

Step 1.2.2

Subtract $2$ from $1$ .

$y = - 1$

Step 2

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

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

$0 + \frac{d}{d x} [- 2 x^{2}]$

Step 2.2

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

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Step 2.2.1

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

$0 - 2 \frac{d}{d x} [x^{2}]$

Step 2.2.2

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

$0 - 2 (2 x)$

Step 2.2.3

Multiply $2$ by $- 2$ .

$0 - 4 x$

Step 2.3

Subtract $4 x$ from $0$ .

$- 4 x$

Step 2.4

Evaluate the derivative at $x = - 1$ .

$- 4 \cdot - 1$

Step 2.5

Multiply $- 4$ by $- 1$ .

$4$

Step 3

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

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Step 3.1

Use the slope $4$ and a given point $(- 1, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = 4 \cdot (x - (- 1))$

Step 3.2

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

$y + 1 = 4 \cdot (x + 1)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $4 \cdot (x + 1)$ .

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Step 3.3.1.1

Rewrite.

$y + 1 = 0 + 0 + 4 \cdot (x + 1)$

Step 3.3.1.2

Simplify by adding zeros.

$y + 1 = 4 \cdot (x + 1)$

Step 3.3.1.3

Apply the distributive property.

$y + 1 = 4 x + 4 \cdot 1$

Step 3.3.1.4

Multiply $4$ by $1$ .

$y + 1 = 4 x + 4$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $1$ from both sides of the equation.

$y = 4 x + 4 - 1$

Step 3.3.2.2

Subtract $1$ from $4$ .

$y = 4 x + 3$

Step 4