Calculus Examples

$f (x) = - x^{2} - x + 4$ at $x = - 1$

Step 1

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

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Step 1.1

Substitute $- 1$ in for $x$ .

$y = - {(- 1)}^{2} - (- 1) + 4$

Step 1.2

Simplify $- {(- 1)}^{2} - (- 1) + 4$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.2.1.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 1.2.1.1.1.1

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

$y = {(- 1)}^{1} {(- 1)}^{2} - (- 1) + 4$

Step 1.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = {(- 1)}^{1 + 2} - (- 1) + 4$

Step 1.2.1.1.2

Add $1$ and $2$ .

$y = {(- 1)}^{3} - (- 1) + 4$

Step 1.2.1.2

Raise $- 1$ to the power of $3$ .

$y = - 1 - (- 1) + 4$

Step 1.2.1.3

Multiply $- 1$ by $- 1$ .

$y = - 1 + 1 + 4$

Step 1.2.2

Simplify by adding numbers.

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Step 1.2.2.1

Add $- 1$ and $1$ .

$y = 0 + 4$

Step 1.2.2.2

Add $0$ and $4$ .

$y = 4$

Step 2

Find the first derivative and evaluate at $x = - 1$ and $y = 4$ to find the slope of the tangent line.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $- 1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $- 1$ by $1$ .

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

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- 2 x - 1 + 0$

Step 2.4.2

Add $- 2 x - 1$ and $0$ .

$- 2 x - 1$

Step 2.5

Evaluate the derivative at $x = - 1$ .

$- 2 \cdot - 1 - 1$

Step 2.6

Simplify.

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Step 2.6.1

Multiply $- 2$ by $- 1$ .

$2 - 1$

Step 2.6.2

Subtract $1$ from $2$ .

$1$

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

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

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

Multiply $x + 1$ by $1$ .

$y - 4 = x + 1$

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

Add $4$ to both sides of the equation.

$y = x + 1 + 4$

Step 3.3.2.2

Add $1$ and $4$ .

$y = x + 5$

Step 4