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 cos(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
\[ \frac{d}{dx}\left[\cos\left(5 x\right)\right] = -\left(5 \sin\left(5 x\right)\right) \]
\[ f(x) = \cos\left(5 x\right) \]
\[ g(x) = 5 x,\quad g'(x) = 5 \]
\[ f'(x) = \left[-\sin\left(u\right)\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = -\left(5 \sin\left(5 x\right)\right) \]
\[ f''(x) = -\left(25 \cos\left(5 x\right)\right) \]Values of f and f' for cos(5x)
| x | f(x) | f'(x) |
|---|---|---|
| -2 | -0.83907153 | -2.7201056 |
| -1 | 0.28366219 | -4.7946214 |
| 0.5 | -0.80114362 | -2.9923607 |
| 1 | 0.28366219 | 4.7946214 |
| 2 | -0.83907153 | 2.7201056 |
| 3 | -0.75968791 | -3.2514392 |
Questions
What is the derivative of cos(5x)?
The derivative of is .
What is the second derivative of cos(5x)?
-(25cos(5x)).
Which rule do you use for cos(5x)?
The chain rule.
What is the slope of cos(5x) at x = 1?
4.7946214.