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
\[ \frac{d}{dx}\left[x^{4} x^{5}\right] = 4 x^{3} x^{5} + x^{4} 5 x^{4} \]
\[ f(x) = x^{4} x^{5} \]
\[ u = x^{4},\quad v = x^{5} \]
\[ u' = 4 x^{3},\quad v' = 5 x^{4} \]
\[ f'(x) = \left(4 x^{3}\right)\left(x^{5}\right) + \left(x^{4}\right)\left(5 x^{4}\right) \]
\[ f'(x) = 4 x^{3} x^{5} + x^{4} 5 x^{4} \]
\[ f''(x) = 12 x^{2} x^{5} + 4 x^{3} 5 x^{4} + 4 x^{3} 5 x^{4} + x^{4} 20 x^{3} \]Values of f and f' for *
| x | f(x) | f'(x) |
|---|---|---|
| -2 | -512 | 2304 |
| -1 | -1 | 9 |
| 0.5 | 0.001953125 | 0.03515625 |
| 1 | 1 | 9 |
| 2 | 512 | 2304 |
| 3 | 19683 | 59049 |
Questions
What is the derivative of * ?
The derivative of is .
What is the second derivative of * ?
12 * * + 4 * * 5 * + 4 * * 5 * + * 20 * .
Which rule do you use for * ?
The product rule.
What is the slope of * at x = 1?
9.