Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Differentiate using the Constant Multiple Rule.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

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 $g (x) = \ln (x^{2})$ .

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

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 $g (x) = x^{2}$ .

Tap for more steps...

Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.4

Differentiate using the Power Rule.

Tap for more steps...

Step 1.4.1

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

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

Step 1.4.2

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

Tap for more steps...

Step 1.4.2.1

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

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

Step 1.4.2.2

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

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

Step 1.4.2.3

Cancel the common factors.

Tap for more steps...

Step 1.4.2.3.1

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

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

Step 1.4.2.3.2

Cancel the common factor.

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

Step 1.4.2.3.3

Rewrite the expression.

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

Step 1.4.3

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

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

Step 1.4.4

Simplify terms.

Tap for more steps...

Step 1.4.4.1

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

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

Step 1.4.4.2

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

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

Step 1.4.4.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 1.4.4.3.1

Cancel the common factor.

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

Step 1.4.4.3.2

Divide $2$ by $1$ .

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

Step 1.4.5

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

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

Step 1.4.6

Multiply $\ln (x^{2})$ by $1$ .

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

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Combine terms.

Tap for more steps...

Step 1.5.2.1

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

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

Step 1.5.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.2.2.1

Cancel the common factor.

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

Step 1.5.2.2.2

Rewrite the expression.

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

Step 1.5.3

Reorder terms.

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

Step 1.5.4

Simplify each term.

Tap for more steps...

Step 1.5.4.1

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

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

Step 1.5.4.2

Expand $\ln (x^{2})$ by moving $2$ outside the logarithm.

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

Step 1.5.4.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.4.3.1

Cancel the common factor.

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

Step 1.5.4.3.2

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

$\ln (x) + 1$

Step 1.6

Evaluate the derivative at $x = - 1$ .

$\ln (- 1) + 1$

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