Derivative of
The derivative of is \frac{5 x^{4} e^{x} - x^{5} e^{x}}{e^{x}^{2}}.
f(x) =
\frac{d}{dx}\left[\frac{x^{5}}{e^{x}}\right] = \frac{5 x^{4} e^{x} - x^{5} e^{x}}{e^{x}^{2}}
Type any function of x; the derivative is computed symbolically in your browser.
How to work it out
- 1
We differentiate with respect to x. The rule that applies is the quotient rule.
- 2
Quotient rule: . Let and .
- 3
Differentiate each piece:
- 4
Put them in the rule:
- 5
Simplify:
f'(x) = \frac{5 x^{4} e^{x} - x^{5} e^{x}}{e^{x}^{2}} - 6
Differentiating once more gives the second derivative, which tells you about curvature:
f''(x) = \frac{\left(20 x^{3} e^{x} + 5 x^{4} e^{x} - \left(5 x^{4} e^{x} + x^{5} e^{x}\right)\right) e^{x}^{2} - \left(5 x^{4} e^{x} - x^{5} e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}}
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\frac{x^{5}}{e^{x}}\right] = \frac{5 x^{4} e^{x} - x^{5} e^{x}}{e^{x}^{2}} \]
\[ f(x) = \frac{x^{5}}{e^{x}} \]
\[ u = x^{5},\quad v = e^{x} \]
\[ u' = 5 x^{4},\quad v' = e^{x} \]
\[ f'(x) = \frac{\left(5 x^{4}\right)\left(e^{x}\right) - \left(x^{5}\right)\left(e^{x}\right)}{\left(e^{x}\right)^{2}} \]
\[ f'(x) = \frac{5 x^{4} e^{x} - x^{5} e^{x}}{e^{x}^{2}} \]
\[ f''(x) = \frac{\left(20 x^{3} e^{x} + 5 x^{4} e^{x} - \left(5 x^{4} e^{x} + x^{5} e^{x}\right)\right) e^{x}^{2} - \left(5 x^{4} e^{x} - x^{5} e^{x}\right) 2 e^{x} e^{x}}{\left(e^{x}^{2}\right)^{2}} \]Values of f and f' for / exp(x)
| x | f(x) | f'(x) |
|---|---|---|
| -2 | -236.4498 | 827.57428 |
| -1 | -2.7182818 | 16.309691 |
| 0.5 | 0.018954083 | 0.17058675 |
| 1 | 0.36787944 | 1.4715178 |
| 2 | 4.3307291 | 6.4960936 |
| 3 | 12.098258 | 8.0655051 |
Questions
What is the derivative of / exp(x)?
The derivative of is \frac{5 x^{4} e^{x} - x^{5} e^{x}}{e^{x}^{2}}.
What is the second derivative of / exp(x)?
((20 * * exp(x) + 5 * * exp(x) - (5 * * exp(x) + * exp(x))) * exp(x)^2 - (5 * * exp(x) - * exp(x)) * 2exp(x) * exp(x)) / (exp(x)^2)^2.
Which rule do you use for / exp(x)?
The quotient rule.
What is the slope of / exp(x) at x = 1?
1.4715178.