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

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

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

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

f(x) =

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

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

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

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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) = -1.

How to work it out

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

LaTeX source of the formulas on this page

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

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

x	f(x)	f'(x)
-2	-0.5	-0.25
-1	-1	-1
0.5	2	-4
1	1	-1
2	0.5	-0.25
3	0.33333333	-0.11111111

Questions

What is the derivative of

x^{2}

x^{3}

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

What is the second derivative of

x^{2}

x^{3}

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

Which rule do you use for

x^{2}

x^{3}

The quotient rule.

What is the slope of

x^{2}

x^{3}

at x = 1?

-1.

Derivative of x2x3\frac{x^{2}}{x^{3}}x3x2​

How to work it out

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

Questions

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

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