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 sin(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
\[ \frac{d}{dx}\left[\sin\left(x^{2} + 1\right)\right] = \cos\left(x^{2} + 1\right) 2 x \]
\[ f(x) = \sin\left(x^{2} + 1\right) \]
\[ g(x) = x^{2} + 1,\quad g'(x) = 2 x \]
\[ f'(x) = \left[\cos\left(u\right)\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \cos\left(x^{2} + 1\right) 2 x \]
\[ f''(x) = -\left(\sin\left(x^{2} + 1\right) 2 x 2 x\right) + 2 \cos\left(x^{2} + 1\right) \]Values of f and f' for sin( + 1)
| x | f(x) | f'(x) |
|---|---|---|
| -2 | -0.95892427 | -1.1346487 |
| -1 | 0.90929743 | 0.83229367 |
| 0.5 | 0.94898462 | 0.31532236 |
| 1 | 0.90929743 | -0.83229367 |
| 2 | -0.95892427 | 1.1346487 |
| 3 | -0.54402111 | -5.0344292 |
Questions
What is the derivative of sin( + 1)?
The derivative of is .
What is the second derivative of sin( + 1)?
-(sin( + 1) * 2x * 2x) + 2cos( + 1).
Which rule do you use for sin( + 1)?
The chain rule.
What is the slope of sin( + 1) at x = 1?
-0.83229367.