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 ; 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
\[ \frac{d}{dx}\left[e^{5 x}\right] = 5 e^{5 x} \]
\[ f(x) = e^{5 x} \]
\[ g(x) = 5 x,\quad g'(x) = 5 \]
\[ f'(x) = \left[e^{u}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = 5 e^{5 x} \]
\[ f''(x) = 25 e^{5 x} \]Values of f and f' for exp(5x)
| x | f(x) | f'(x) |
|---|---|---|
| -2 | 0.00004539993 | 0.00022699965 |
| -1 | 0.006737947 | 0.033689735 |
| 0.5 | 12.182494 | 60.91247 |
| 1 | 148.41316 | 742.0658 |
| 2 | 22026.466 | 110132.33 |
| 3 | 3269017.4 | 16345087 |
Questions
What is the derivative of exp(5x)?
The derivative of is .
What is the second derivative of exp(5x)?
25exp(5x).
Which rule do you use for exp(5x)?
The chain rule.
What is the slope of exp(5x) at x = 1?
742.0658.