Calculus Examples

$y = \frac{25}{x^{2}}$ at the point $(5, 1)$

Step 1

Find the first derivative and evaluate at $x = 5$ and $y = 1$ to find the slope of the tangent line.

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Step 1.1

Since $25$ is constant with respect to $x$ , the derivative of $\frac{25}{x^{2}}$ with respect to $x$ is $25 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$25 \frac{d}{d x} [\frac{1}{x^{2}}]$

Step 1.2

Apply basic rules of exponents.

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Step 1.2.1

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

$25 \frac{d}{d x} [{(x^{2})}^{- 1}]$

Step 1.2.2

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

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Step 1.2.2.1

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

$25 \frac{d}{d x} [x^{2 \cdot - 1}]$

Step 1.2.2.2

Multiply $2$ by $- 1$ .

$25 \frac{d}{d x} [x^{- 2}]$

Step 1.3

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

$25 (- 2 x^{- 3})$

Step 1.4

Multiply $- 2$ by $25$ .

$- 50 x^{- 3}$

Step 1.5

Simplify.

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Step 1.5.1

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

$- 50 \frac{1}{x^{3}}$

Step 1.5.2

Combine terms.

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Step 1.5.2.1

Combine $- 50$ and $\frac{1}{x^{3}}$ .

$\frac{- 50}{x^{3}}$

Step 1.5.2.2

Move the negative in front of the fraction.

$- \frac{50}{x^{3}}$

Step 1.6

Evaluate the derivative at $x = 5$ .

$- \frac{50}{{(5)}^{3}}$

Step 1.7

Simplify.

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Step 1.7.1

Raise $5$ to the power of $3$ .

$- \frac{50}{125}$

Step 1.7.2

Cancel the common factor of $50$ and $125$ .

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Step 1.7.2.1

Factor $25$ out of $50$ .

$- \frac{25 (2)}{125}$

Step 1.7.2.2

Cancel the common factors.

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Step 1.7.2.2.1

Factor $25$ out of $125$ .

$- \frac{25 \cdot 2}{25 \cdot 5}$

Step 1.7.2.2.2

Cancel the common factor.

$- \frac{25 \cdot 2}{25 \cdot 5}$

Step 1.7.2.2.3

Rewrite the expression.

$- \frac{2}{5}$

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 $- \frac{2}{5}$ and a given point $(5, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = - \frac{2}{5} \cdot (x - (5))$

Step 2.2

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

$y - 1 = - \frac{2}{5} \cdot (x - 5)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- \frac{2}{5} \cdot (x - 5)$ .

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Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 - \frac{2}{5} \cdot (x - 5)$

Step 2.3.1.2

Simplify terms.

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Step 2.3.1.2.1

Apply the distributive property.

$y - 1 = - \frac{2}{5} x - \frac{2}{5} \cdot - 5$

Step 2.3.1.2.2

Combine $x$ and $\frac{2}{5}$ .

$y - 1 = - \frac{x \cdot 2}{5} - \frac{2}{5} \cdot - 5$

Step 2.3.1.2.3

Cancel the common factor of $5$ .

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Step 2.3.1.2.3.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$y - 1 = - \frac{x \cdot 2}{5} + \frac{- 2}{5} \cdot - 5$

Step 2.3.1.2.3.2

Factor $5$ out of $- 5$ .

$y - 1 = - \frac{x \cdot 2}{5} + \frac{- 2}{5} \cdot (5 (- 1))$

Step 2.3.1.2.3.3

Cancel the common factor.

$y - 1 = - \frac{x \cdot 2}{5} + \frac{- 2}{5} \cdot (5 \cdot - 1)$

Step 2.3.1.2.3.4

Rewrite the expression.

$y - 1 = - \frac{x \cdot 2}{5} - 2 \cdot - 1$

Step 2.3.1.2.4

Multiply $- 2$ by $- 1$ .

$y - 1 = - \frac{x \cdot 2}{5} + 2$

Step 2.3.1.3

Move $2$ to the left of $x$ .

$y - 1 = - \frac{2 x}{5} + 2$

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.

$y = - \frac{2 x}{5} + 2 + 1$

Step 2.3.2.2

Add $2$ and $1$ .

$y = - \frac{2 x}{5} + 3$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

Reorder terms.

$y = - (\frac{2}{5} x) + 3$

Step 2.3.3.2

Remove parentheses.

$y = - \frac{2}{5} x + 3$

Step 3