Calculus Examples

$f (x) = x^{2} + \ln (x)$

Step 1

Find the derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{2} + \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [\ln (x)]$ .

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

Step 1.1.2

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

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

Step 1.2

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

$2 x + \frac{1}{x}$

Step 2

Set the derivative equal to $0$ then solve the equation $2 x + \frac{1}{x} = 0$ .

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Step 2.1

Find the LCD of the terms in the equation.

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Step 2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x, 1$

Step 2.1.2

The LCM of one and any expression is the expression.

$x$

Step 2.2

Multiply each term in $2 x + \frac{1}{x} = 0$ by $x$ to eliminate the fractions.

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Step 2.2.1

Multiply each term in $2 x + \frac{1}{x} = 0$ by $x$ .

$2 x \cdot x + \frac{1}{x} x = 0 x$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.2.1.1.1

Move $x$ .

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

Step 2.2.2.1.1.2

Multiply $x$ by $x$ .

$2 x^{2} + \frac{1}{x} x = 0 x$

Step 2.2.2.1.2

Cancel the common factor of $x$ .

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Step 2.2.2.1.2.1

Cancel the common factor.

$2 x^{2} + \frac{1}{x} x = 0 x$

Step 2.2.2.1.2.2

Rewrite the expression.

$2 x^{2} + 1 = 0 x$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply $0$ by $x$ .

$2 x^{2} + 1 = 0$

Step 2.3

Solve the equation.

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Step 2.3.1

Subtract $1$ from both sides of the equation.

$2 x^{2} = - 1$

Step 2.3.2

Divide each term in $2 x^{2} = - 1$ by $2$ and simplify.

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Step 2.3.2.1

Divide each term in $2 x^{2} = - 1$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{- 1}{2}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $2$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{- 1}{2}$

Step 2.3.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 1}{2}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{2}$

Step 2.3.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- \frac{1}{2}}$

Step 2.3.4

Simplify $\pm \sqrt{- \frac{1}{2}}$ .

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Step 2.3.4.1

Rewrite $- \frac{1}{2}$ as $i^{2} \frac{1^{2}}{2}$ .

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Step 2.3.4.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1}{2}}$

Step 2.3.4.1.2

Rewrite $1$ as $1^{2}$ .

$x = \pm \sqrt{i^{2} \frac{1^{2}}{2}}$

Step 2.3.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{1^{2}}{2}}$

Step 2.3.4.3

One to any power is one.

$x = \pm i \sqrt{\frac{1}{2}}$

Step 2.3.4.4

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{1}}{\sqrt{2}}$

Step 2.3.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt{2}}$

Step 2.3.4.6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 2.3.4.7

Combine and simplify the denominator.

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Step 2.3.4.7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.3.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.3.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.3.4.7.4

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

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.3.4.7.5

Add $1$ and $1$ .

$x = \pm i \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.3.4.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.3.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.3.4.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm i \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.3.4.7.6.3

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

$x = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.3.4.7.6.4

Cancel the common factor of $2$ .

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Step 2.3.4.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.3.4.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{\sqrt{2}}{2^{1}}$

Step 2.3.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2}}{2}$

Step 2.3.4.8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$x = \pm \frac{i \sqrt{2}}{2}$

Step 2.3.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.3.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{i \sqrt{2}}{2}$

Step 2.3.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{i \sqrt{2}}{2}$

Step 2.3.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}$

Step 3

A tangent line cannot be found at an imaginary point. The point at $x =$ does not exist on the real coordinate system.

A tangent cannot be found from the root

Step 4

There are no horizontal tangent lines on the function $f (x) = x^{2} + \ln (x)$ .

No horizontal tangent lines

Step 5