Derivative of $x^{2} x^{3}$

Q: What is the derivative of x^2 * x^3?

The derivative of x^2 x^3 is 2 x x^3 + x^2 3 x^2.

Q: What is the second derivative of x^2 * x^3?

2 * x^3 + 2x * 3 * x^2 + 2x * 3 * x^2 + x^2 * 6x.

The derivative of $x^{2} x^{3}$ is $2 x x^{3} + x^{2} 3 x^{2}$ .

f(x) =

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

Type any function of x; the derivative is computed symbolically in your browser.

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

Explain it step by step in chat Make a short video of it

Graph of f(x) = x^2 * x^3 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 5.

How to work it out

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

LaTeX source of the formulas on this page

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

Values of f and f' for $x^{2}$ * $x^{3}$

x	f(x)	f'(x)
-2	-32	80
-1	-1	5
0.5	0.03125	0.3125
1	1	5
2	32	80
3	243	405

Questions

What is the derivative of

x^{2}

x^{3}

The derivative of $x^{2} x^{3}$ is $2 x x^{3} + x^{2} 3 x^{2}$ .

What is the second derivative of

x^{2}

x^{3}

2 * $x^{3}$ + 2x * 3 * $x^{2}$ + 2x * 3 * $x^{2}$ + $x^{2}$ * 6x.

Which rule do you use for

x^{2}

x^{3}

The product rule.

What is the slope of

x^{2}

x^{3}

at x = 1?

Derivative of x2x3x^{2} x^{3}x2x3

How to work it out

Values of f and f' for x2x^{2}x2 * x3x^{3}x3

Questions

Related derivatives

Derivative of $x^{2} x^{3}$

Values of f and f' for $x^{2}$ * $x^{3}$