Calculus Examples

$y = 4 x^{3} \sqrt{x}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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

$\frac{d}{d x} [4 x^{3} x^{\frac{1}{2}}]$

Step 1.2

Multiply $x^{3}$ by $x^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 1.2.1

Move $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [4 (x^{\frac{1}{2}} x^{3})]$

Step 1.2.2

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

$\frac{d}{d x} [4 x^{\frac{1}{2} + 3}]$

Step 1.2.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{d}{d x} [4 x^{\frac{1}{2} + 3 \cdot \frac{2}{2}}]$

Step 1.2.4

Combine $3$ and $\frac{2}{2}$ .

$\frac{d}{d x} [4 x^{\frac{1}{2} + \frac{3 \cdot 2}{2}}]$

Step 1.2.5

Combine the numerators over the common denominator.

$\frac{d}{d x} [4 x^{\frac{1 + 3 \cdot 2}{2}}]$

Step 1.2.6

Simplify the numerator.

Tap for more steps...

Step 1.2.6.1

Multiply $3$ by $2$ .

$\frac{d}{d x} [4 x^{\frac{1 + 6}{2}}]$

Step 1.2.6.2

Add $1$ and $6$ .

$\frac{d}{d x} [4 x^{\frac{7}{2}}]$

Step 1.3

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{\frac{7}{2}}$ with respect to $x$ is $4 \frac{d}{d x} [x^{\frac{7}{2}}]$ .

$4 \frac{d}{d x} [x^{\frac{7}{2}}]$

Step 1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{2}$ .

$4 (\frac{7}{2} x^{\frac{7}{2} - 1})$

Step 1.5

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

$4 (\frac{7}{2} x^{\frac{7}{2} - 1 \cdot \frac{2}{2}})$

Step 1.6

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

$4 (\frac{7}{2} x^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}})$

Step 1.7

Combine the numerators over the common denominator.

$4 (\frac{7}{2} x^{\frac{7 - 1 \cdot 2}{2}})$

Step 1.8

Simplify the numerator.

Tap for more steps...

Step 1.8.1

Multiply $- 1$ by $2$ .

$4 (\frac{7}{2} x^{\frac{7 - 2}{2}})$

Step 1.8.2

Subtract $2$ from $7$ .

$4 (\frac{7}{2} x^{\frac{5}{2}})$

Step 1.9

Combine $\frac{7}{2}$ and $x^{\frac{5}{2}}$ .

$4 \frac{7 x^{\frac{5}{2}}}{2}$

Step 1.10

Combine $4$ and $\frac{7 x^{\frac{5}{2}}}{2}$ .

$\frac{4 (7 x^{\frac{5}{2}})}{2}$

Step 1.11

Multiply $7$ by $4$ .

$\frac{28 x^{\frac{5}{2}}}{2}$

Step 1.12

Factor $2$ out of $28 x^{\frac{5}{2}}$ .

$\frac{2 (14 x^{\frac{5}{2}})}{2}$

Step 1.13

Cancel the common factors.

Tap for more steps...

Step 1.13.1

Factor $2$ out of $2$ .

$\frac{2 (14 x^{\frac{5}{2}})}{2 (1)}$

Step 1.13.2

Cancel the common factor.

$\frac{2 (14 x^{\frac{5}{2}})}{2 \cdot 1}$

Step 1.13.3

Rewrite the expression.

$\frac{14 x^{\frac{5}{2}}}{1}$

Step 1.13.4

Divide $14 x^{\frac{5}{2}}$ by $1$ .

$f' (x) = 14 x^{\frac{5}{2}}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x^{\frac{5}{2}}$ with respect to $x$ is $14 \frac{d}{d x} [x^{\frac{5}{2}}]$ .

$14 \frac{d}{d x} [x^{\frac{5}{2}}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$14 (\frac{5}{2} x^{\frac{5}{2} - 1})$

Step 2.3

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

$14 (\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}})$

Step 2.4

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

$14 (\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}})$

Step 2.5

Combine the numerators over the common denominator.

$14 (\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}})$

Step 2.6

Simplify the numerator.

Tap for more steps...

Step 2.6.1

Multiply $- 1$ by $2$ .

$14 (\frac{5}{2} x^{\frac{5 - 2}{2}})$

Step 2.6.2

Subtract $2$ from $5$ .

$14 (\frac{5}{2} x^{\frac{3}{2}})$

Step 2.7

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$14 \frac{5 x^{\frac{3}{2}}}{2}$

Step 2.8

Combine $14$ and $\frac{5 x^{\frac{3}{2}}}{2}$ .

$\frac{14 (5 x^{\frac{3}{2}})}{2}$

Step 2.9

Multiply $5$ by $14$ .

$\frac{70 x^{\frac{3}{2}}}{2}$

Step 2.10

Factor $2$ out of $70 x^{\frac{3}{2}}$ .

$\frac{2 (35 x^{\frac{3}{2}})}{2}$

Step 2.11

Cancel the common factors.

Tap for more steps...

Step 2.11.1

Factor $2$ out of $2$ .

$\frac{2 (35 x^{\frac{3}{2}})}{2 (1)}$

Step 2.11.2

Cancel the common factor.

$\frac{2 (35 x^{\frac{3}{2}})}{2 \cdot 1}$

Step 2.11.3

Rewrite the expression.

$\frac{35 x^{\frac{3}{2}}}{1}$

Step 2.11.4

Divide $35 x^{\frac{3}{2}}$ by $1$ .

$f'' (x) = 35 x^{\frac{3}{2}}$