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

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

The derivative of x^2x^4 is 2 x x^4 - x^2 4 x^3≤ft(x^4)^2.

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)) * (x^4)^2 - (2x * x^4 - x^2 * 4 * x^3) * 2 * x^4 * 4 * x^3) / ((x^4)^2)^2.

The derivative of $\frac{x^{2}}{x^{4}}$ is $\frac{2 x x^{4} - x^{2} 4 x^{3}}{\left(x^{4}\right)^{2}}$ .

f(x) =

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

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

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

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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) = -2.

How to work it out

1
We differentiate $f(x) = \frac{x^{2}}{x^{4}}$ with respect to x. The rule that applies is the quotient rule.
$f(x) = \frac{x^{2}}{x^{4}}$
2
Quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}$ . 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:
$f'(x) = \frac{\left(2 x\right)\left(x^{4}\right) - \left(x^{2}\right)\left(4 x^{3}\right)}{\left(x^{4}\right)^{2}}$
5
Simplify:
$f'(x) = \frac{2 x x^{4} - x^{2} 4 x^{3}}{\left(x^{4}\right)^{2}}$
6
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = \frac{\left(2 x^{4} + 2 x 4 x^{3} - \left(2 x 4 x^{3} + x^{2} 12 x^{2}\right)\right) \left(x^{4}\right)^{2} - \left(2 x x^{4} - x^{2} 4 x^{3}\right) 2 x^{4} 4 x^{3}}{\left(\left(x^{4}\right)^{2}\right)^{2}}$

LaTeX source of the formulas on this page

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

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

x	f(x)	f'(x)
-2	0.25	0.25
-1	1	2
0.5	4	-16
1	1	-2
2	0.25	-0.25
3	0.11111111	-0.074074074

Questions

What is the derivative of

x^{2}

x^{4}

The derivative of $\frac{x^{2}}{x^{4}}$ is $\frac{2 x x^{4} - x^{2} 4 x^{3}}{\left(x^{4}\right)^{2}}$ .

What is the second derivative of

x^{2}

x^{4}

((2 * $x^{4}$ + 2x * 4 * $x^{3}$ - (2x * 4 * $x^{3}$ + $x^{2}$ * 12 * $x^{2}$ )) * ( $x^{4}$ )^2 - (2x * $x^{4}$ - $x^{2}$ * 4 * $x^{3}$ ) * 2 * $x^{4}$ * 4 * $x^{3}$ ) / (( $x^{4}$ )^2)^2.

Which rule do you use for

x^{2}

x^{4}

The quotient rule.

What is the slope of

x^{2}

x^{4}

at x = 1?

-2.

Derivative of x2x4\frac{x^{2}}{x^{4}}x4x2​

How to work it out

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

Questions

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

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