Calculus Examples

$f (x) = - \frac{1}{4} x^{2}$ ;, $(- 2, - 1)$

Step 1

Find the first derivative and evaluate at $x = - 2$ and $y = - 1$ to find the slope of the tangent line.

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Step 1.1

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{1}{4} x^{2}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [x^{2}]$ .

$- \frac{1}{4} \frac{d}{d x} [x^{2}]$

Step 1.2

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

$- \frac{1}{4} (2 x)$

Step 1.3

Simplify terms.

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Step 1.3.1

Multiply $2$ by $- 1$ .

$- 2 (\frac{1}{4}) x$

Step 1.3.2

Combine $- 2$ and $\frac{1}{4}$ .

$\frac{- 2}{4} x$

Step 1.3.3

Combine $\frac{- 2}{4}$ and $x$ .

$\frac{- 2 x}{4}$

Step 1.3.4

Cancel the common factor of $- 2$ and $4$ .

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Step 1.3.4.1

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x)}{4}$

Step 1.3.4.2

Cancel the common factors.

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Step 1.3.4.2.1

Factor $2$ out of $4$ .

$\frac{2 (- x)}{2 (2)}$

Step 1.3.4.2.2

Cancel the common factor.

$\frac{2 (- x)}{2 \cdot 2}$

Step 1.3.4.2.3

Rewrite the expression.

$\frac{- x}{2}$

Step 1.3.5

Move the negative in front of the fraction.

$- \frac{x}{2}$

Step 1.4

Evaluate the derivative at $x = - 2$ .

$- \frac{- 2}{2}$

Step 1.5

Simplify.

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Step 1.5.1

Divide $- 2$ by $2$ .

$- - 1$

Step 1.5.2

Multiply $- 1$ by $- 1$ .

$1$

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 $(- 2, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = 1 \cdot (x - (- 2))$

Step 2.2

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

$y + 1 = 1 \cdot (x + 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $x + 2$ by $1$ .

$y + 1 = x + 2$

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 $1$ from both sides of the equation.

$y = x + 2 - 1$

Step 2.3.2.2

Subtract $1$ from $2$ .

$y = x + 1$

Step 3