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
\[ \frac{d}{dx}\left[x^{3} x^{5}\right] = 3 x^{2} x^{5} + x^{3} 5 x^{4} \]
\[ f(x) = x^{3} x^{5} \]
\[ u = x^{3},\quad v = x^{5} \]
\[ u' = 3 x^{2},\quad v' = 5 x^{4} \]
\[ f'(x) = \left(3 x^{2}\right)\left(x^{5}\right) + \left(x^{3}\right)\left(5 x^{4}\right) \]
\[ f'(x) = 3 x^{2} x^{5} + x^{3} 5 x^{4} \]
\[ f''(x) = 6 x x^{5} + 3 x^{2} 5 x^{4} + 3 x^{2} 5 x^{4} + x^{3} 20 x^{3} \]Values of f and f' for *
| x | f(x) | f'(x) |
|---|---|---|
| -2 | 256 | -1024 |
| -1 | 1 | -8 |
| 0.5 | 0.00390625 | 0.0625 |
| 1 | 1 | 8 |
| 2 | 256 | 1024 |
| 3 | 6561 | 17496 |
Questions
What is the derivative of * ?
The derivative of is .
What is the second derivative of * ?
6x * + 3 * * 5 * + 3 * * 5 * + * 20 * .
Which rule do you use for * ?
The product rule.
What is the slope of * at x = 1?
8.