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 chain rule.
- 2
Chain rule: differentiate the outer function, keep the inside, then multiply by the derivative of the inside. The inside is .
- 3
The outer function is sqrt(u); its derivative evaluated at g(x), times g'(x):
- 4
Simplify:
- 5
Differentiating once more gives the second derivative, which tells you about curvature:
LaTeX source of the formulas on this page
\[ \frac{d}{dx}\left[\sqrt{x^{3}}\right] = \frac{3 x^{2}}{2 \sqrt{x^{3}}} \]
\[ f(x) = \sqrt{x^{3}} \]
\[ g(x) = x^{3},\quad g'(x) = 3 x^{2} \]
\[ f'(x) = \left[\frac{1}{2 \sqrt{u}}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{3 x^{2}}{2 \sqrt{x^{3}}} \]
\[ f''(x) = \frac{6 x 2 \sqrt{x^{3}} - 3 x^{2} 2 \frac{3 x^{2}}{2 \sqrt{x^{3}}}}{\left(2 \sqrt{x^{3}}\right)^{2}} \]Values of f and f' for sqrt()
| x | f(x) | f'(x) |
|---|---|---|
| 0.5 | 0.35355339 | 1.0606602 |
| 1 | 1 | 1.5 |
| 2 | 2.8284271 | 2.1213203 |
| 3 | 5.1961524 | 2.5980762 |
Questions
What is the derivative of sqrt()?
The derivative of is .
What is the second derivative of sqrt()?
(6x * 2sqrt() - 3 * * 2 * 3 * / (2sqrt())) / (2sqrt())^2.
Which rule do you use for sqrt()?
The chain rule.
What is the slope of sqrt() at x = 1?
1.5.