Calculus Examples

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

Step 1

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 1.1

Apply basic rules of exponents.

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Step 1.1.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 1.1.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 1.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.1.2.2

Multiply $2$ by $- 1$ .

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

Step 1.2

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

$- 2 x^{- 3}$

Step 1.3

Simplify.

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Step 1.3.1

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

$- 2 \frac{1}{x^{3}}$

Step 1.3.2

Combine terms.

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Step 1.3.2.1

Combine $- 2$ and $\frac{1}{x^{3}}$ .

$\frac{- 2}{x^{3}}$

Step 1.3.2.2

Move the negative in front of the fraction.

$- \frac{2}{x^{3}}$

Step 1.4

Evaluate the derivative at $x = 1$ .

$- \frac{2}{{(1)}^{3}}$

Step 1.5

Simplify.

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Step 1.5.1

One to any power is one.

$- \frac{2}{1}$

Step 1.5.2

Divide $2$ by $1$ .

$- 1 \cdot 2$

Step 1.5.3

Multiply $- 1$ by $2$ .

$- 2$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 2 \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 - 2 \cdot (x - 1)$

Step 2.3.1.2

Simplify by adding zeros.

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

Step 2.3.1.3

Apply the distributive property.

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

Step 2.3.1.4

Multiply $- 2$ by $- 1$ .

$y - 1 = - 2 x + 2$

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

Add $1$ to both sides of the equation.

$y = - 2 x + 2 + 1$

Step 2.3.2.2

Add $2$ and $1$ .

$y = - 2 x + 3$

Step 3