Derivative of $x^{3} \sqrt{x}$

Q: What is the derivative of x^3 * sqrt(x)?

The derivative of x^3 √(x) is 3 x^2 √(x) + x^3 (1)/(2 √(x)).

Q: What is the second derivative of x^3 * sqrt(x)?

6x * sqrt(x) + 3 * x^2 * 1 / (2sqrt(x)) + 3 * x^2 * 1 / (2sqrt(x)) + x^3 * -(2 * 1 / (2sqrt(x))) / (2sqrt(x))^2.

The derivative of $x^{3} \sqrt{x}$ is $3 x^{2} \sqrt{x} + x^{3} \frac{1}{2 \sqrt{x}}$ .

f(x) =

\frac{d}{dx}\left[x^{3} \sqrt{x}\right] = 3 x^{2} \sqrt{x} + x^{3} \frac{1}{2 \sqrt{x}}

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\frac{d}{dx}\left[x^{3} \sqrt{x}\right] = 3 x^{2} \sqrt{x} + x^{3} \frac{1}{2 \sqrt{x}}

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Graph of f(x) = x^3 * sqrt(x) and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.5.

How to work it out

1
We differentiate $f(x) = x^{3} \sqrt{x}$ with respect to x. The rule that applies is the product rule.
$f(x) = x^{3} \sqrt{x}$
2
Product rule: $(uv)' = u'v + uv'$ . Let $u = x^{3}$ and $v = \sqrt{x}$ .
$u = x^{3},\quad v = \sqrt{x}$
3
Differentiate each piece:
$u' = 3 x^{2},\quad v' = \frac{1}{2 \sqrt{x}}$
4
Put them in the rule:
$f'(x) = \left(3 x^{2}\right)\left(\sqrt{x}\right) + \left(x^{3}\right)\left(\frac{1}{2 \sqrt{x}}\right)$
5
Simplify:
$f'(x) = 3 x^{2} \sqrt{x} + x^{3} \frac{1}{2 \sqrt{x}}$
6
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = 6 x \sqrt{x} + 3 x^{2} \frac{1}{2 \sqrt{x}} + 3 x^{2} \frac{1}{2 \sqrt{x}} + x^{3} \frac{-\left(2 \frac{1}{2 \sqrt{x}}\right)}{\left(2 \sqrt{x}\right)^{2}}$

LaTeX source of the formulas on this page

\[ \frac{d}{dx}\left[x^{3} \sqrt{x}\right] = 3 x^{2} \sqrt{x} + x^{3} \frac{1}{2 \sqrt{x}} \]
\[ f(x) = x^{3} \sqrt{x} \]
\[ u = x^{3},\quad v = \sqrt{x} \]
\[ u' = 3 x^{2},\quad v' = \frac{1}{2 \sqrt{x}} \]
\[ f'(x) = \left(3 x^{2}\right)\left(\sqrt{x}\right) + \left(x^{3}\right)\left(\frac{1}{2 \sqrt{x}}\right) \]
\[ f'(x) = 3 x^{2} \sqrt{x} + x^{3} \frac{1}{2 \sqrt{x}} \]
\[ f''(x) = 6 x \sqrt{x} + 3 x^{2} \frac{1}{2 \sqrt{x}} + 3 x^{2} \frac{1}{2 \sqrt{x}} + x^{3} \frac{-\left(2 \frac{1}{2 \sqrt{x}}\right)}{\left(2 \sqrt{x}\right)^{2}} \]

Values of f and f' for $x^{3}$ * sqrt(x)

x	f(x)	f'(x)
0.5	0.088388348	0.61871843
1	1	3.5
2	11.313708	19.79899
3	46.765372	54.5596

Questions

What is the derivative of

x^{3}

* sqrt(x)?

The derivative of $x^{3} \sqrt{x}$ is $3 x^{2} \sqrt{x} + x^{3} \frac{1}{2 \sqrt{x}}$ .

What is the second derivative of

x^{3}

* sqrt(x)?

6x * sqrt(x) + 3 * $x^{2}$ * 1 / (2sqrt(x)) + 3 * $x^{2}$ * 1 / (2sqrt(x)) + $x^{3}$ * -(2 * 1 / (2sqrt(x))) / (2sqrt(x))^2.

Which rule do you use for

x^{3}

* sqrt(x)?

The product rule.

What is the slope of

x^{3}

* sqrt(x) at x = 1?

3.5.

Derivative of x3xx^{3} \sqrt{x}x3x​

How to work it out

Values of f and f' for x3x^{3}x3 * sqrt(x)

Questions

Related derivatives

Derivative of $x^{3} \sqrt{x}$

Values of f and f' for $x^{3}$ * sqrt(x)