Calculus Examples

f (x) = \frac{1}{2} x ln (x^{4})

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

$(- 1, 0)$

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