Calculus Examples

$f (x) = \frac{1}{2} x \ln (x^{4})$ , $(- 1, 0)$

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 Constant Multiple Rule.

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Step 1.1.1

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

$\frac{d}{d x} [\frac{x}{2} \cdot \ln (x^{4})]$

Step 1.1.2

Combine $\frac{x}{2}$ and $\ln (x^{4})$ .

$\frac{d}{d x} [\frac{x \ln (x^{4})}{2}]$

Step 1.1.3

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x \ln (x^{4})}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x \ln (x^{4})]$ .

$\frac{1}{2} \frac{d}{d x} [x \ln (x^{4})]$

Step 1.2

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 $g (x) = \ln (x^{4})$ .

$\frac{1}{2} (x \frac{d}{d x} [\ln (x^{4})] + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.3

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 $g (x) = x^{4}$ .

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Step 1.3.1

To apply the Chain Rule, set $u$ as $x^{4}$ .

$\frac{1}{2} (x (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{4}]) + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.3.2

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

$\frac{1}{2} (x (\frac{1}{u} \frac{d}{d x} [x^{4}]) + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.3.3

Replace all occurrences of $u$ with $x^{4}$ .

$\frac{1}{2} (x (\frac{1}{x^{4}} \frac{d}{d x} [x^{4}]) + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4

Differentiate using the Power Rule.

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Step 1.4.1

Combine $\frac{1}{x^{4}}$ and $x$ .

$\frac{1}{2} (\frac{x}{x^{4}} \frac{d}{d x} [x^{4}] + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.2

Cancel the common factor of $x$ and $x^{4}$ .

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Step 1.4.2.1

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

$\frac{1}{2} (\frac{x^{1}}{x^{4}} \frac{d}{d x} [x^{4}] + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.2.2

Factor $x$ out of $x^{1}$ .

$\frac{1}{2} (\frac{x \cdot 1}{x^{4}} \frac{d}{d x} [x^{4}] + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.2.3

Cancel the common factors.

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Step 1.4.2.3.1

Factor $x$ out of $x^{4}$ .

$\frac{1}{2} (\frac{x \cdot 1}{x \cdot x^{3}} \frac{d}{d x} [x^{4}] + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.2.3.2

Cancel the common factor.

$\frac{1}{2} (\frac{x \cdot 1}{x \cdot x^{3}} \frac{d}{d x} [x^{4}] + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.2.3.3

Rewrite the expression.

$\frac{1}{2} (\frac{1}{x^{3}} \frac{d}{d x} [x^{4}] + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{2} (\frac{1}{x^{3}} (4 x^{3}) + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.4

Simplify terms.

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Step 1.4.4.1

Combine $4$ and $\frac{1}{x^{3}}$ .

$\frac{1}{2} (\frac{4}{x^{3}} x^{3} + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.4.2

Combine $\frac{4}{x^{3}}$ and $x^{3}$ .

$\frac{1}{2} (\frac{4 x^{3}}{x^{3}} + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.4.3

Cancel the common factor of $x^{3}$ .

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Step 1.4.4.3.1

Cancel the common factor.

$\frac{1}{2} (\frac{4 x^{3}}{x^{3}} + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.4.3.2

Divide $4$ by $1$ .

$\frac{1}{2} (4 + \ln (x^{4}) \frac{d}{d x} [x])$

Step 1.4.5

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

$\frac{1}{2} (4 + \ln (x^{4}) \cdot 1)$

Step 1.4.6

Multiply $\ln (x^{4})$ by $1$ .

$\frac{1}{2} (4 + \ln (x^{4}))$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$\frac{1}{2} \cdot 4 + \frac{1}{2} \ln (x^{4})$

Step 1.5.2

Combine terms.

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Step 1.5.2.1

Combine $\frac{1}{2}$ and $4$ .

$\frac{4}{2} + \frac{1}{2} \ln (x^{4})$

Step 1.5.2.2

Cancel the common factor of $4$ and $2$ .

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Step 1.5.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2} + \frac{1}{2} \ln (x^{4})$

Step 1.5.2.2.2

Cancel the common factors.

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Step 1.5.2.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 2}{2 (1)} + \frac{1}{2} \ln (x^{4})$

Step 1.5.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 1} + \frac{1}{2} \ln (x^{4})$

Step 1.5.2.2.2.3

Rewrite the expression.

$\frac{2}{1} + \frac{1}{2} \ln (x^{4})$

Step 1.5.2.2.2.4

Divide $2$ by $1$ .

$2 + \frac{1}{2} \ln (x^{4})$

Step 1.5.3

Reorder terms.

$\frac{1}{2} \ln (x^{4}) + 2$

Step 1.5.4

Simplify each term.

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Step 1.5.4.1

Combine $\frac{1}{2}$ and $\ln (x^{4})$ .

$\frac{\ln (x^{4})}{2} + 2$

Step 1.5.4.2

Expand $\ln (x^{4})$ by moving $4$ outside the logarithm.

$\frac{4 \ln (x)}{2} + 2$

Step 1.5.4.3

Cancel the common factor of $4$ and $2$ .

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Step 1.5.4.3.1

Factor $2$ out of $4 \ln (x)$ .

$\frac{2 (2 \ln (x))}{2} + 2$

Step 1.5.4.3.2

Cancel the common factors.

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Step 1.5.4.3.2.1

Factor $2$ out of $2$ .

$\frac{2 (2 \ln (x))}{2 (1)} + 2$

Step 1.5.4.3.2.2

Cancel the common factor.

$\frac{2 (2 \ln (x))}{2 \cdot 1} + 2$

Step 1.5.4.3.2.3

Rewrite the expression.

$\frac{2 \ln (x)}{1} + 2$

Step 1.5.4.3.2.4

Divide $2 \ln (x)$ by $1$ .

$2 \ln (x) + 2$

Step 1.6

Evaluate the derivative at $x = - 1$ .

$2 \ln (- 1) + 2$

Step 1.7

The natural logarithm of a negative number is undefined.

Undefined

Step 2

The slope of the line is undefined, which means that it is perpendicular to the x-axis at $x = - 1$ .

$x = - 1$

Step 3