Calculus Examples

$y = x (16 x^{- 2} + 2)$ ; $x = - 2$

Step 1

Find the corresponding $y$ -value to $x = - 2$ .

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Step 1.1

Substitute $- 2$ in for $x$ .

$y = (- 2) (16 {(- 2)}^{- 2} + 2)$

Step 1.2

Solve for $y$ .

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Step 1.2.1

Multiply $- 2$ by $16 {(- 2)}^{- 2} + 2$ .

$y = - 2 (16 {(- 2)}^{- 2} + 2)$

Step 1.2.2

Simplify $- 2 (16 {(- 2)}^{- 2} + 2)$ .

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Step 1.2.2.1

Simplify each term.

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Step 1.2.2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y = - 2 (16 \frac{1}{{(- 2)}^{2}} + 2)$

Step 1.2.2.1.2

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

$y = - 2 (16 (\frac{1}{4}) + 2)$

Step 1.2.2.1.3

Cancel the common factor of $4$ .

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Step 1.2.2.1.3.1

Factor $4$ out of $16$ .

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

Step 1.2.2.1.3.2

Cancel the common factor.

$y = - 2 (4 \cdot 4 \frac{1}{4} + 2)$

Step 1.2.2.1.3.3

Rewrite the expression.

$y = - 2 (4 + 2)$

Step 1.2.2.2

Simplify the expression.

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Step 1.2.2.2.1

Add $4$ and $2$ .

$y = - 2 \cdot 6$

Step 1.2.2.2.2

Multiply $- 2$ by $6$ .

$y = - 12$

Step 2

Find the first derivative and evaluate at $x = - 2$ and $y = - 12$ to find the slope of the tangent line.

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Step 2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = 16 x^{- 2} + 2$ .

$x \frac{d}{d x} [16 x^{- 2} + 2] + (16 x^{- 2} + 2) \frac{d}{d x} [x]$

Step 2.2

Differentiate.

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Step 2.2.1

By the Sum Rule, the derivative of $16 x^{- 2} + 2$ with respect to $x$ is $\frac{d}{d x} [16 x^{- 2}] + \frac{d}{d x} [2]$ .

$x (\frac{d}{d x} [16 x^{- 2}] + \frac{d}{d x} [2]) + (16 x^{- 2} + 2) \frac{d}{d x} [x]$

Step 2.2.2

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{- 2}$ with respect to $x$ is $16 \frac{d}{d x} [x^{- 2}]$ .

$x (16 \frac{d}{d x} [x^{- 2}] + \frac{d}{d x} [2]) + (16 x^{- 2} + 2) \frac{d}{d x} [x]$

Step 2.2.3

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

$x (16 (- 2 x^{- 3}) + \frac{d}{d x} [2]) + (16 x^{- 2} + 2) \frac{d}{d x} [x]$

Step 2.2.4

Multiply $- 2$ by $16$ .

$x (- 32 x^{- 3} + \frac{d}{d x} [2]) + (16 x^{- 2} + 2) \frac{d}{d x} [x]$

Step 2.2.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$x (- 32 x^{- 3} + 0) + (16 x^{- 2} + 2) \frac{d}{d x} [x]$

Step 2.2.6

Add $- 32 x^{- 3}$ and $0$ .

$x (- 32 x^{- 3}) + (16 x^{- 2} + 2) \frac{d}{d x} [x]$

Step 2.3

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

$- 32 (x^{1} x^{- 3}) + (16 x^{- 2} + 2) \frac{d}{d x} [x]$

Step 2.4

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

$- 32 x^{1 - 3} + (16 x^{- 2} + 2) \frac{d}{d x} [x]$

Step 2.5

Subtract $3$ from $1$ .

$- 32 x^{- 2} + (16 x^{- 2} + 2) \frac{d}{d x} [x]$

Step 2.6

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

$- 32 x^{- 2} + (16 x^{- 2} + 2) \cdot 1$

Step 2.7

Simplify terms.

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Step 2.7.1

Multiply $16 x^{- 2} + 2$ by $1$ .

$- 32 x^{- 2} + 16 x^{- 2} + 2$

Step 2.7.2

Add $- 32 x^{- 2}$ and $16 x^{- 2}$ .

$- 16 x^{- 2} + 2$

Step 2.7.3

Simplify each term.

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Step 2.7.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 16 \frac{1}{x^{2}} + 2$

Step 2.7.3.2

Combine $- 16$ and $\frac{1}{x^{2}}$ .

$\frac{- 16}{x^{2}} + 2$

Step 2.7.3.3

Move the negative in front of the fraction.

$- \frac{16}{x^{2}} + 2$

Step 2.8

Evaluate the derivative at $x = - 2$ .

$- \frac{16}{{(- 2)}^{2}} + 2$

Step 2.9

Simplify.

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Step 2.9.1

Simplify each term.

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Step 2.9.1.1

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

$- \frac{16}{4} + 2$

Step 2.9.1.2

Divide $16$ by $4$ .

$- 1 \cdot 4 + 2$

Step 2.9.1.3

Multiply $- 1$ by $4$ .

$- 4 + 2$

Step 2.9.2

Add $- 4$ and $2$ .

$- 2$

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

$y - (- 12) = - 2 \cdot (x - (- 2))$

Step 3.2

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

$y + 12 = - 2 \cdot (x + 2)$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Simplify $- 2 \cdot (x + 2)$ .

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Step 3.3.1.1

Rewrite.

$y + 12 = 0 + 0 - 2 \cdot (x + 2)$

Step 3.3.1.2

Simplify by adding zeros.

$y + 12 = - 2 \cdot (x + 2)$

Step 3.3.1.3

Apply the distributive property.

$y + 12 = - 2 x - 2 \cdot 2$

Step 3.3.1.4

Multiply $- 2$ by $2$ .

$y + 12 = - 2 x - 4$

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 $12$ from both sides of the equation.

$y = - 2 x - 4 - 12$

Step 3.3.2.2

Subtract $12$ from $- 4$ .

$y = - 2 x - 16$

Step 4