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
\[ \frac{d}{dx}\left[\cos\left(1 - x\right)\right] = \sin\left(1 - x\right) \]
\[ f(x) = \cos\left(1 - x\right) \]
\[ g(x) = 1 - x,\quad g'(x) = -1 \]
\[ f'(x) = \left[-\sin\left(u\right)\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \sin\left(1 - x\right) \]
\[ f''(x) = -\cos\left(1 - x\right) \]Values of f and f' for cos(1 - x)
| x | f(x) | f'(x) |
|---|---|---|
| -2 | -0.9899925 | 0.14112001 |
| -1 | -0.41614684 | 0.90929743 |
| 0.5 | 0.87758256 | 0.47942554 |
| 1 | 1 | 0 |
| 2 | 0.54030231 | -0.84147098 |
| 3 | -0.41614684 | -0.90929743 |
Questions
What is the derivative of cos(1 - x)?
The derivative of is .
What is the second derivative of cos(1 - x)?
-cos(1 - x).
Which rule do you use for cos(1 - x)?
The chain rule.
What is the slope of cos(1 - x) at x = 1?
0.