Calculus Examples

$y = k \sqrt{x} + x \sqrt{k}$

Step 1

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

$\frac{d}{d k} [k \sqrt{x}] + \frac{d}{d k} [x \sqrt{k}]$

Step 2

Evaluate $\frac{d}{d k} [k \sqrt{x}]$ .

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Step 2.1

Since $\sqrt{x}$ is constant with respect to $k$ , the derivative of $k \sqrt{x}$ with respect to $k$ is $\sqrt{x} \frac{d}{d k} [k]$ .

$\sqrt{x} \frac{d}{d k} [k] + \frac{d}{d k} [x \sqrt{k}]$

Step 2.2

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

$\sqrt{x} \cdot 1 + \frac{d}{d k} [x \sqrt{k}]$

Step 2.3

Multiply $\sqrt{x}$ by $1$ .

$\sqrt{x} + \frac{d}{d k} [x \sqrt{k}]$

Step 3

Evaluate $\frac{d}{d k} [x \sqrt{k}]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$\sqrt{x} + x (\frac{1}{2} k^{\frac{1}{2} - 1})$

Step 3.4

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

$\sqrt{x} + x (\frac{1}{2} k^{\frac{1}{2} - 1 \cdot \frac{2}{2}})$

Step 3.5

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

$\sqrt{x} + x (\frac{1}{2} k^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.6

Combine the numerators over the common denominator.

$\sqrt{x} + x (\frac{1}{2} k^{\frac{1 - 1 \cdot 2}{2}})$

Step 3.7

Simplify the numerator.

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Step 3.7.1

Multiply $- 1$ by $2$ .

$\sqrt{x} + x (\frac{1}{2} k^{\frac{1 - 2}{2}})$

Step 3.7.2

Subtract $2$ from $1$ .

$\sqrt{x} + x (\frac{1}{2} k^{\frac{- 1}{2}})$

Step 3.8

Move the negative in front of the fraction.

$\sqrt{x} + x (\frac{1}{2} k^{- \frac{1}{2}})$

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Move $k^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4

Reorder terms.

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