Derivative of $x^{2} x^{4}$

Q: What is the derivative of x^2 * x^4?

The derivative of x^2 x^4 is 2 x x^4 + x^2 4 x^3.

Q: What is the second derivative of x^2 * x^4?

2 * x^4 + 2x * 4 * x^3 + 2x * 4 * x^3 + x^2 * 12 * x^2.

The derivative of $x^{2} x^{4}$ is $2 x x^{4} + x^{2} 4 x^{3}$ .

f(x) =

\frac{d}{dx}\left[x^{2} x^{4}\right] = 2 x x^{4} + x^{2} 4 x^{3}

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

\frac{d}{dx}\left[x^{2} x^{4}\right] = 2 x x^{4} + x^{2} 4 x^{3}

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Graph of f(x) = x^2 * x^4 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 6.

How to work it out

1
We differentiate $f(x) = x^{2} x^{4}$ with respect to x. The rule that applies is the product rule.
$f(x) = x^{2} x^{4}$
2
Product rule: $(uv)' = u'v + uv'$ . Let $u = x^{2}$ and $v = x^{4}$ .
$u = x^{2},\quad v = x^{4}$
3
Differentiate each piece:
$u' = 2 x,\quad v' = 4 x^{3}$
4
Put them in the rule:
$f'(x) = \left(2 x\right)\left(x^{4}\right) + \left(x^{2}\right)\left(4 x^{3}\right)$
5
Simplify:
$f'(x) = 2 x x^{4} + x^{2} 4 x^{3}$
6
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = 2 x^{4} + 2 x 4 x^{3} + 2 x 4 x^{3} + x^{2} 12 x^{2}$

LaTeX source of the formulas on this page

\[ \frac{d}{dx}\left[x^{2} x^{4}\right] = 2 x x^{4} + x^{2} 4 x^{3} \]
\[ f(x) = x^{2} x^{4} \]
\[ u = x^{2},\quad v = x^{4} \]
\[ u' = 2 x,\quad v' = 4 x^{3} \]
\[ f'(x) = \left(2 x\right)\left(x^{4}\right) + \left(x^{2}\right)\left(4 x^{3}\right) \]
\[ f'(x) = 2 x x^{4} + x^{2} 4 x^{3} \]
\[ f''(x) = 2 x^{4} + 2 x 4 x^{3} + 2 x 4 x^{3} + x^{2} 12 x^{2} \]

Values of f and f' for $x^{2}$ * $x^{4}$

x	f(x)	f'(x)
-2	64	-192
-1	1	-6
0.5	0.015625	0.1875
1	1	6
2	64	192
3	729	1458

Questions

What is the derivative of

x^{2}

x^{4}

The derivative of $x^{2} x^{4}$ is $2 x x^{4} + x^{2} 4 x^{3}$ .

What is the second derivative of

x^{2}

x^{4}

2 * $x^{4}$ + 2x * 4 * $x^{3}$ + 2x * 4 * $x^{3}$ + $x^{2}$ * 12 * $x^{2}$ .

Which rule do you use for

x^{2}

x^{4}

The product rule.

What is the slope of

x^{2}

x^{4}

at x = 1?

Derivative of x2x4x^{2} x^{4}x2x4

How to work it out

Values of f and f' for x2x^{2}x2 * x4x^{4}x4

Questions

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Derivative of $x^{2} x^{4}$

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