Calculus Examples

$y = \ln (x^{\frac{9}{2}})$ ; $(1, 0)$

Step 1

Find the first derivative and evaluate at $x = 1$ and $y = 0$ to find the slope of the tangent line.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{\frac{9}{2}}$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $x^{\frac{9}{2}}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{\frac{9}{2}}]$

Step 1.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [x^{\frac{9}{2}}]$

Step 1.1.3

Replace all occurrences of $u$ with $x^{\frac{9}{2}}$ .

$\frac{1}{x^{\frac{9}{2}}} \frac{d}{d x} [x^{\frac{9}{2}}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{9}{2}$ .

$\frac{1}{x^{\frac{9}{2}}} (\frac{9}{2} x^{\frac{9}{2} - 1})$

Step 1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{x^{\frac{9}{2}}} (\frac{9}{2} x^{\frac{9}{2} - 1 \cdot \frac{2}{2}})$

Step 1.4

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

$\frac{1}{x^{\frac{9}{2}}} (\frac{9}{2} x^{\frac{9}{2} + \frac{- 1 \cdot 2}{2}})$

Step 1.5

Combine the numerators over the common denominator.

$\frac{1}{x^{\frac{9}{2}}} (\frac{9}{2} x^{\frac{9 - 1 \cdot 2}{2}})$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{x^{\frac{9}{2}}} (\frac{9}{2} x^{\frac{9 - 2}{2}})$

Step 1.6.2

Subtract $2$ from $9$ .

$\frac{1}{x^{\frac{9}{2}}} (\frac{9}{2} x^{\frac{7}{2}})$

Step 1.7

Combine $\frac{9}{2}$ and $x^{\frac{7}{2}}$ .

$\frac{1}{x^{\frac{9}{2}}} \cdot \frac{9 x^{\frac{7}{2}}}{2}$

Step 1.8

Multiply $\frac{1}{x^{\frac{9}{2}}}$ by $\frac{9 x^{\frac{7}{2}}}{2}$ .

$\frac{9 x^{\frac{7}{2}}}{x^{\frac{9}{2}} \cdot 2}$

Step 1.9

Simplify the expression.

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Step 1.9.1

Move $2$ to the left of $x^{\frac{9}{2}}$ .

$\frac{9 x^{\frac{7}{2}}}{2 \cdot x^{\frac{9}{2}}}$

Step 1.9.2

Move $x^{\frac{7}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{9}{2 x^{\frac{9}{2}} x^{- \frac{7}{2}}}$

Step 1.10

Simplify the denominator.

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Step 1.10.1

Multiply $x^{\frac{9}{2}}$ by $x^{- \frac{7}{2}}$ by adding the exponents.

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Step 1.10.1.1

Move $x^{- \frac{7}{2}}$ .

$\frac{9}{2 (x^{- \frac{7}{2}} x^{\frac{9}{2}})}$

Step 1.10.1.2

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

$\frac{9}{2 x^{- \frac{7}{2} + \frac{9}{2}}}$

Step 1.10.1.3

Combine the numerators over the common denominator.

$\frac{9}{2 x^{\frac{- 7 + 9}{2}}}$

Step 1.10.1.4

Add $- 7$ and $9$ .

$\frac{9}{2 x^{\frac{2}{2}}}$

Step 1.10.1.5

Divide $2$ by $2$ .

$\frac{9}{2 x^{1}}$

Step 1.10.2

Simplify $2 x^{1}$ .

$\frac{9}{2 x}$

Step 1.11

Evaluate the derivative at $x = 1$ .

$\frac{9}{2 (1)}$

Step 1.12

Multiply $2$ by $1$ .

$\frac{9}{2}$

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 $\frac{9}{2}$ and a given point $(1, 0)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = \frac{9}{2} \cdot (x - (1))$

Step 2.2

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

$y + 0 = \frac{9}{2} \cdot (x - 1)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

$y = \frac{9}{2} \cdot (x - 1)$

Step 2.3.2

Simplify $\frac{9}{2} \cdot (x - 1)$ .

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Step 2.3.2.1

Apply the distributive property.

$y = \frac{9}{2} x + \frac{9}{2} \cdot - 1$

Step 2.3.2.2

Combine $\frac{9}{2}$ and $x$ .

$y = \frac{9 x}{2} + \frac{9}{2} \cdot - 1$

Step 2.3.2.3

Multiply $\frac{9}{2} \cdot - 1$ .

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Step 2.3.2.3.1

Combine $\frac{9}{2}$ and $- 1$ .

$y = \frac{9 x}{2} + \frac{9 \cdot - 1}{2}$

Step 2.3.2.3.2

Multiply $9$ by $- 1$ .

$y = \frac{9 x}{2} + \frac{- 9}{2}$

Step 2.3.2.4

Move the negative in front of the fraction.

$y = \frac{9 x}{2} - \frac{9}{2}$

Step 2.3.3

Reorder terms.

$y = \frac{9}{2} x - \frac{9}{2}$

Step 3