Derivative of $\frac{x^{5}}{x^{2}}$

Q: What is the derivative of x^5 / x^2?

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

Q: What is the second derivative of x^5 / x^2?

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

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

f(x) =

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

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

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

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Graph of f(x) = x^5 / x^2 and its derivative. The dashed line is the tangent at x = 1; its slope is f'(1) = 3.

How to work it out

1
We differentiate $f(x) = \frac{x^{5}}{x^{2}}$ with respect to x. The rule that applies is the quotient rule.
$f(x) = \frac{x^{5}}{x^{2}}$
2
Quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^{2}}$ . Let $u = x^{5}$ and $v = x^{2}$ .
$u = x^{5},\quad v = x^{2}$
3
Differentiate each piece:
$u' = 5 x^{4},\quad v' = 2 x$
4
Put them in the rule:
$f'(x) = \frac{\left(5 x^{4}\right)\left(x^{2}\right) - \left(x^{5}\right)\left(2 x\right)}{\left(x^{2}\right)^{2}}$
5
Simplify:
$f'(x) = \frac{5 x^{4} x^{2} - x^{5} 2 x}{\left(x^{2}\right)^{2}}$
6
Differentiating once more gives the second derivative, which tells you about curvature:
$f''(x) = \frac{\left(20 x^{3} x^{2} + 5 x^{4} 2 x - \left(5 x^{4} 2 x + 2 x^{5}\right)\right) \left(x^{2}\right)^{2} - \left(5 x^{4} x^{2} - x^{5} 2 x\right) 2 x^{2} 2 x}{\left(\left(x^{2}\right)^{2}\right)^{2}}$

LaTeX source of the formulas on this page

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

Values of f and f' for $x^{5}$ / $x^{2}$

x	f(x)	f'(x)
-2	-8	12
-1	-1	3
0.5	0.125	0.75
1	1	3
2	8	12
3	27	27

Questions

What is the derivative of

x^{5}

x^{2}

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

What is the second derivative of

x^{5}

x^{2}

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

Which rule do you use for

x^{5}

x^{2}

The quotient rule.

What is the slope of

x^{5}

x^{2}

at x = 1?

Derivative of x5x2\frac{x^{5}}{x^{2}}x2x5​

How to work it out

Values of f and f' for x5x^{5}x5 / x2x^{2}x2

Questions

Related derivatives

Derivative of $\frac{x^{5}}{x^{2}}$

Values of f and f' for $x^{5}$ / $x^{2}$