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
\[ \frac{d}{dx}\left[\cos\left(x^{2} + 1\right)\right] = -\left(\sin\left(x^{2} + 1\right) 2 x\right) \]
\[ f(x) = \cos\left(x^{2} + 1\right) \]
\[ g(x) = x^{2} + 1,\quad g'(x) = 2 x \]
\[ f'(x) = \left[-\sin\left(u\right)\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = -\left(\sin\left(x^{2} + 1\right) 2 x\right) \]
\[ f''(x) = -\left(\cos\left(x^{2} + 1\right) 2 x 2 x + 2 \sin\left(x^{2} + 1\right)\right) \]Values of f and f' for cos( + 1)
| x | f(x) | f'(x) |
|---|---|---|
| -2 | 0.28366219 | -3.8356971 |
| -1 | -0.41614684 | 1.8185949 |
| 0.5 | 0.31532236 | -0.94898462 |
| 1 | -0.41614684 | -1.8185949 |
| 2 | 0.28366219 | 3.8356971 |
| 3 | -0.83907153 | 3.2641267 |
Questions
What is the derivative of cos( + 1)?
The derivative of is .
What is the second derivative of cos( + 1)?
-(cos( + 1) * 2x * 2x + 2sin( + 1)).
Which rule do you use for cos( + 1)?
The chain rule.
What is the slope of cos( + 1) at x = 1?
-1.8185949.