Calculus Examples

$y = \ln (x^{2})$ , $(2, \ln (4))$

Step 1

Find the first derivative and evaluate at $x = 2$ and $y = \ln (4)$ 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 $g (x) = x^{2}$ .

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

Differentiate using the Power Rule.

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Step 1.2.1

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

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

Step 1.2.2

Simplify terms.

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.2.3

Cancel the common factor of $x$ and $x^{2}$ .

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Step 1.2.2.3.1

Factor $x$ out of $2 x$ .

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

Step 1.2.2.3.2

Cancel the common factors.

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Step 1.2.2.3.2.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot 2}{x \cdot x}$

Step 1.2.2.3.2.2

Cancel the common factor.

$\frac{x \cdot 2}{x \cdot x}$

Step 1.2.2.3.2.3

Rewrite the expression.

$\frac{2}{x}$

Step 1.3

Evaluate the derivative at $x = 2$ .

$\frac{2}{2}$

Step 1.4

Divide $2$ by $2$ .

$1$

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

$y - (\ln (4)) = 1 \cdot (x - (2))$

Step 2.2

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

$y - \ln (4) = 1 \cdot (x - 2)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $x - 2$ by $1$ .

$y - \ln (4) = x - 2$

Step 2.3.2

Add $\ln (4)$ to both sides of the equation.

$y = x - 2 + \ln (4)$

Step 3