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
\[ \frac{d}{dx}\left[\sqrt{5 x}\right] = \frac{5}{2 \sqrt{5 x}} \]
\[ f(x) = \sqrt{5 x} \]
\[ g(x) = 5 x,\quad g'(x) = 5 \]
\[ f'(x) = \left[\frac{1}{2 \sqrt{u}}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{5}{2 \sqrt{5 x}} \]
\[ f''(x) = \frac{-\left(10 \frac{5}{2 \sqrt{5 x}}\right)}{\left(2 \sqrt{5 x}\right)^{2}} \]Values of f and f' for sqrt(5x)
| x | f(x) | f'(x) |
|---|---|---|
| 0.5 | 1.5811388 | 1.5811388 |
| 1 | 2.236068 | 1.118034 |
| 2 | 3.1622777 | 0.79056942 |
| 3 | 3.8729833 | 0.64549722 |
Questions
What is the derivative of sqrt(5x)?
The derivative of is .
What is the second derivative of sqrt(5x)?
-(10 * 5 / (2sqrt(5x))) / (2sqrt(5x))^2.
Which rule do you use for sqrt(5x)?
The chain rule.
What is the slope of sqrt(5x) at x = 1?
1.118034.