Calculus Examples

$y = 16 x^{\frac{1}{2}} + 4 x$

Step 1

By the Sum Rule, the derivative of $16 x^{\frac{1}{2}} + 4 x$ with respect to $x$ is $\frac{d}{d x} [16 x^{\frac{1}{2}}] + \frac{d}{d x} [4 x]$ .

$\frac{d}{d x} [16 x^{\frac{1}{2}}] + \frac{d}{d x} [4 x]$

Step 2

Evaluate $\frac{d}{d x} [16 x^{\frac{1}{2}}]$ .

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Step 2.1

Since $16$ is constant with respect to $x$ , the derivative of $16 x^{\frac{1}{2}}$ with respect to $x$ is $16 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$16 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [4 x]$

Step 2.2

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

$16 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [4 x]$

Step 2.3

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

$16 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [4 x]$

Step 2.4

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

$16 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [4 x]$

Step 2.5

Combine the numerators over the common denominator.

$16 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [4 x]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $2$ .

$16 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [4 x]$

Step 2.6.2

Subtract $2$ from $1$ .

$16 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [4 x]$

Step 2.7

Move the negative in front of the fraction.

$16 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [4 x]$

Step 2.8

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

$16 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [4 x]$

Step 2.9

Combine $16$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{16 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [4 x]$

Step 2.10

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

$\frac{16}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [4 x]$

Step 2.11

Factor $2$ out of $16$ .

$\frac{2 \cdot 8}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [4 x]$

Step 2.12

Cancel the common factors.

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Step 2.12.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$\frac{2 \cdot 8}{2 (x^{\frac{1}{2}})} + \frac{d}{d x} [4 x]$

Step 2.12.2

Cancel the common factor.

$\frac{2 \cdot 8}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [4 x]$

Step 2.12.3

Rewrite the expression.

$\frac{8}{x^{\frac{1}{2}}} + \frac{d}{d x} [4 x]$

Step 3

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$\frac{8}{x^{\frac{1}{2}}} + 4 \frac{d}{d x} [x]$

Step 3.2

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

$\frac{8}{x^{\frac{1}{2}}} + 4 \cdot 1$

Step 3.3

Multiply $4$ by $1$ .

$\frac{8}{x^{\frac{1}{2}}} + 4$