Derivative of $\frac{x^{3}}{x^{5}}$

Q: What is the derivative of x^3 / x^5?

The derivative of x^3x^5 is 3 x^2 x^5 - x^3 5 x^4≤ft(x^5)^2.

Q: What is the second derivative of x^3 / x^5?

((6x * x^5 + 3 * x^2 * 5 * x^4 - (3 * x^2 * 5 * x^4 + x^3 * 20 * x^3)) * (x^5)^2 - (3 * x^2 * x^5 - x^3 * 5 * x^4) * 2 * x^5 * 5 * x^4) / ((x^5)^2)^2.

The derivative of $\frac{x^{3}}{x^{5}}$ is $\frac{3 x^{2} x^{5} - x^{3} 5 x^{4}}{\left(x^{5}\right)^{2}}$ .

f(x) =

\frac{d}{dx}\left[\frac{x^{3}}{x^{5}}\right] = \frac{3 x^{2} x^{5} - x^{3} 5 x^{4}}{\left(x^{5}\right)^{2}}

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

\frac{d}{dx}\left[\frac{x^{3}}{x^{5}}\right] = \frac{3 x^{2} x^{5} - x^{3} 5 x^{4}}{\left(x^{5}\right)^{2}}

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

Graph of f(x) = x^3 / x^5 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 $f(x) = \frac{x^{3}}{x^{5}}$ with respect to x. The rule that applies is the quotient rule.
$f(x) = \frac{x^{3}}{x^{5}}$
2
Quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}$ . Let $u = x^{3}$ and $v = x^{5}$ .
$u = x^{3},\quad v = x^{5}$
3
Differentiate each piece:
$u' = 3 x^{2},\quad v' = 5 x^{4}$
4
Put them in the rule:
$f'(x) = \frac{\left(3 x^{2}\right)\left(x^{5}\right) - \left(x^{3}\right)\left(5 x^{4}\right)}{\left(x^{5}\right)^{2}}$
5
Simplify:
$f'(x) = \frac{3 x^{2} x^{5} - x^{3} 5 x^{4}}{\left(x^{5}\right)^{2}}$
6
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = \frac{\left(6 x x^{5} + 3 x^{2} 5 x^{4} - \left(3 x^{2} 5 x^{4} + x^{3} 20 x^{3}\right)\right) \left(x^{5}\right)^{2} - \left(3 x^{2} x^{5} - x^{3} 5 x^{4}\right) 2 x^{5} 5 x^{4}}{\left(\left(x^{5}\right)^{2}\right)^{2}}$

LaTeX source of the formulas on this page

\[ \frac{d}{dx}\left[\frac{x^{3}}{x^{5}}\right] = \frac{3 x^{2} x^{5} - x^{3} 5 x^{4}}{\left(x^{5}\right)^{2}} \]
\[ f(x) = \frac{x^{3}}{x^{5}} \]
\[ u = x^{3},\quad v = x^{5} \]
\[ u' = 3 x^{2},\quad v' = 5 x^{4} \]
\[ f'(x) = \frac{\left(3 x^{2}\right)\left(x^{5}\right) - \left(x^{3}\right)\left(5 x^{4}\right)}{\left(x^{5}\right)^{2}} \]
\[ f'(x) = \frac{3 x^{2} x^{5} - x^{3} 5 x^{4}}{\left(x^{5}\right)^{2}} \]
\[ f''(x) = \frac{\left(6 x x^{5} + 3 x^{2} 5 x^{4} - \left(3 x^{2} 5 x^{4} + x^{3} 20 x^{3}\right)\right) \left(x^{5}\right)^{2} - \left(3 x^{2} x^{5} - x^{3} 5 x^{4}\right) 2 x^{5} 5 x^{4}}{\left(\left(x^{5}\right)^{2}\right)^{2}} \]

Values of f and f' for $x^{3}$ / $x^{5}$

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^{3}

x^{5}

The derivative of $\frac{x^{3}}{x^{5}}$ is $\frac{3 x^{2} x^{5} - x^{3} 5 x^{4}}{\left(x^{5}\right)^{2}}$ .

What is the second derivative of

x^{3}

x^{5}

((6x * $x^{5}$ + 3 * $x^{2}$ * 5 * $x^{4}$ - (3 * $x^{2}$ * 5 * $x^{4}$ + $x^{3}$ * 20 * $x^{3}$ )) * ( $x^{5}$ )^2 - (3 * $x^{2}$ * $x^{5}$ - $x^{3}$ * 5 * $x^{4}$ ) * 2 * $x^{5}$ * 5 * $x^{4}$ ) / (( $x^{5}$ )^2)^2.

Which rule do you use for

x^{3}

x^{5}

The quotient rule.

What is the slope of

x^{3}

x^{5}

at x = 1?

-2.

Derivative of x3x5\frac{x^{3}}{x^{5}}x5x3​

How to work it out

Values of f and f' for x3x^{3}x3 / x5x^{5}x5

Questions

Related derivatives

Derivative of $\frac{x^{3}}{x^{5}}$

Values of f and f' for $x^{3}$ / $x^{5}$