Derivative of $\frac{x^{4}}{x^{5}}$

Q: What is the derivative of x^4 / x^5?

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

Q: What is the second derivative of x^4 / x^5?

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

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

f(x) =

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

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Graph of f(x) = x^4 / x^5 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 $f(x) = \frac{x^{4}}{x^{5}}$ with respect to x. The rule that applies is the quotient rule.
$f(x) = \frac{x^{4}}{x^{5}}$
2
Quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}$ . Let $u = x^{4}$ and $v = x^{5}$ .
$u = x^{4},\quad v = x^{5}$
3
Differentiate each piece:
$u' = 4 x^{3},\quad v' = 5 x^{4}$
4
Put them in the rule:
$f'(x) = \frac{\left(4 x^{3}\right)\left(x^{5}\right) - \left(x^{4}\right)\left(5 x^{4}\right)}{\left(x^{5}\right)^{2}}$
5
Simplify:
$f'(x) = \frac{4 x^{3} x^{5} - x^{4} 5 x^{4}}{\left(x^{5}\right)^{2}}$
6
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = \frac{\left(12 x^{2} x^{5} + 4 x^{3} 5 x^{4} - \left(4 x^{3} 5 x^{4} + x^{4} 20 x^{3}\right)\right) \left(x^{5}\right)^{2} - \left(4 x^{3} x^{5} - x^{4} 5 x^{4}\right) 2 x^{5} 5 x^{4}}{\left(\left(x^{5}\right)^{2}\right)^{2}}$

LaTeX source of the formulas on this page

\[ \frac{d}{dx}\left[\frac{x^{4}}{x^{5}}\right] = \frac{4 x^{3} x^{5} - x^{4} 5 x^{4}}{\left(x^{5}\right)^{2}} \]
\[ f(x) = \frac{x^{4}}{x^{5}} \]
\[ u = x^{4},\quad v = x^{5} \]
\[ u' = 4 x^{3},\quad v' = 5 x^{4} \]
\[ f'(x) = \frac{\left(4 x^{3}\right)\left(x^{5}\right) - \left(x^{4}\right)\left(5 x^{4}\right)}{\left(x^{5}\right)^{2}} \]
\[ f'(x) = \frac{4 x^{3} x^{5} - x^{4} 5 x^{4}}{\left(x^{5}\right)^{2}} \]
\[ f''(x) = \frac{\left(12 x^{2} x^{5} + 4 x^{3} 5 x^{4} - \left(4 x^{3} 5 x^{4} + x^{4} 20 x^{3}\right)\right) \left(x^{5}\right)^{2} - \left(4 x^{3} x^{5} - x^{4} 5 x^{4}\right) 2 x^{5} 5 x^{4}}{\left(\left(x^{5}\right)^{2}\right)^{2}} \]

Values of f and f' for $x^{4}$ / $x^{5}$

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^{4}

x^{5}

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

What is the second derivative of

x^{4}

x^{5}

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

Which rule do you use for

x^{4}

x^{5}

The quotient rule.

What is the slope of

x^{4}

x^{5}

at x = 1?

-1.

Derivative of x4x5\frac{x^{4}}{x^{5}}x5x4​

How to work it out

Values of f and f' for x4x^{4}x4 / x5x^{5}x5

Questions

Related derivatives

Derivative of $\frac{x^{4}}{x^{5}}$

Values of f and f' for $x^{4}$ / $x^{5}$