Calculus Examples

$y = x \sqrt{x}$

Step 1

Set $y$ as a function of $x$ .

$f (x) = x \sqrt{x}$

Step 2

Find the derivative.

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Step 2.1

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

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

Step 2.2

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 2.2.1

Multiply $x$ by $x^{\frac{1}{2}}$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.2.3

Combine the numerators over the common denominator.

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

Step 2.2.4

Add $2$ and $1$ .

$\frac{d}{d x} [x^{\frac{3}{2}}]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{3}{2} x^{\frac{3}{2} - 1}$

Step 2.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}}$

Step 2.5

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

$\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}}$

Step 2.6

Combine the numerators over the common denominator.

$\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $2$ .

$\frac{3}{2} x^{\frac{3 - 2}{2}}$

Step 2.7.2

Subtract $2$ from $3$ .

$\frac{3}{2} x^{\frac{1}{2}}$

Step 2.8

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$\frac{3 x^{\frac{1}{2}}}{2}$

Step 3

Set the derivative equal to $0$ then solve the equation $\frac{3 x^{\frac{1}{2}}}{2} = 0$ .

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Step 3.1

Set the numerator equal to zero.

$3 x^{\frac{1}{2}} = 0$

Step 3.2

Solve the equation for $x$ .

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Step 3.2.1

Divide each term in $3 x^{\frac{1}{2}} = 0$ by $3$ and simplify.

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Step 3.2.1.1

Divide each term in $3 x^{\frac{1}{2}} = 0$ by $3$ .

$\frac{3 x^{\frac{1}{2}}}{3} = \frac{0}{3}$

Step 3.2.1.2

Simplify the left side.

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Step 3.2.1.2.1

Cancel the common factor.

$\frac{3 x^{\frac{1}{2}}}{3} = \frac{0}{3}$

Step 3.2.1.2.2

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

$x^{\frac{1}{2}} = \frac{0}{3}$

Step 3.2.1.3

Simplify the right side.

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Step 3.2.1.3.1

Divide $0$ by $3$ .

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

Step 3.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 3.2.3

Simplify the exponent.

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Step 3.2.3.1

Simplify the left side.

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Step 3.2.3.1.1

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

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Step 3.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 3.2.3.1.1.1.1

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

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

Step 3.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 3.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 3.2.3.1.1.2

Simplify.

$x = 0^{2}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 4

Solve the original function $f (x) = x \sqrt{x}$ at $x = 0$ .

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Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = (0) \sqrt{0}$

Step 4.2

Simplify the result.

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Step 4.2.1

Remove parentheses.

$f (0) = (0) \sqrt{0}$

Step 4.2.2

Rewrite $0$ as $0^{2}$ .

$f (0) = 0 \sqrt{0^{2}}$

Step 4.2.3

Pull terms out from under the radical, assuming positive real numbers.

$f (0) = 0 \cdot 0$

Step 4.2.4

Multiply $0$ by $0$ .

$f (0) = 0$

Step 4.2.5

The final answer is $0$ .

$0$

Step 5

The horizontal tangent line on function $f (x) = x \sqrt{x}$ is $y = 0$ .

$y = 0$

Step 6