Derivative of
The derivative of is .
f(x) =
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 product rule.
- 2
Product rule: . Let and .
- 3
Differentiate each piece:
- 4
Put them in the rule:
- 5
Simplify:
- 6
Differentiating once more gives the second derivative, which tells you about curvature:
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[x^{5} e^{x}\right] = 5 x^{4} e^{x} + x^{5} e^{x} \]
\[ f(x) = x^{5} e^{x} \]
\[ u = x^{5},\quad v = e^{x} \]
\[ u' = 5 x^{4},\quad v' = e^{x} \]
\[ f'(x) = \left(5 x^{4}\right)\left(e^{x}\right) + \left(x^{5}\right)\left(e^{x}\right) \]
\[ f'(x) = 5 x^{4} e^{x} + x^{5} e^{x} \]
\[ f''(x) = 20 x^{3} e^{x} + 5 x^{4} e^{x} + 5 x^{4} e^{x} + x^{5} e^{x} \]Values of f and f' for * exp(x)
| x | f(x) | f'(x) |
|---|---|---|
| -2 | -4.3307291 | 6.4960936 |
| -1 | -0.36787944 | 1.4715178 |
| 0.5 | 0.05152254 | 0.56674794 |
| 1 | 2.7182818 | 16.309691 |
| 2 | 236.4498 | 827.57428 |
| 3 | 4880.7855 | 13015.428 |
Questions
What is the derivative of * exp(x)?
The derivative of is .
What is the second derivative of * exp(x)?
20 * * exp(x) + 5 * * exp(x) + 5 * * exp(x) + * exp(x).
Which rule do you use for * exp(x)?
The product rule.
What is the slope of * exp(x) at x = 1?
16.309691.