Calculus Examples

$y = x + \sqrt{x}$

Step 1

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

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

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x + x^{\frac{1}{2}})$

Step 3

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Differentiate.

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Step 4.1.1

By the Sum Rule, the derivative of $x + x^{\frac{1}{2}}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [x^{\frac{1}{2}}]$ .

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

Step 4.1.2

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $- 1$ by $2$ .

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

Step 4.2.5.2

Subtract $2$ from $1$ .

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

Step 4.2.6

Move the negative in front of the fraction.

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

Step 4.3

Simplify.

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Step 4.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4.3.2

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

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

Step 5

Reform the equation by setting the left side equal to the right side.

$y' = 1 + \frac{1}{2 x^{\frac{1}{2}}}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 1 + \frac{1}{2 x^{\frac{1}{2}}}$