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 tan(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
\[ \frac{d}{dx}\left[\tan\left(-x\right)\right] = \frac{-1}{\cos\left(-x\right)^{2}} \]
\[ f(x) = \tan\left(-x\right) \]
\[ g(x) = -x,\quad g'(x) = -1 \]
\[ f'(x) = \left[\frac{1}{\cos\left(u\right)^{2}}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{-1}{\cos\left(-x\right)^{2}} \]
\[ f''(x) = \frac{2 \cos\left(-x\right) \sin\left(-x\right)}{\left(\cos\left(-x\right)^{2}\right)^{2}} \]Values of f and f' for tan(-x)
| x | f(x) | f'(x) |
|---|---|---|
| -2 | -2.1850399 | -5.7743992 |
| -1 | 1.5574077 | -3.4255188 |
| 0.5 | -0.54630249 | -1.2984464 |
| 1 | -1.5574077 | -3.4255188 |
| 2 | 2.1850399 | -5.7743992 |
| 3 | 0.14254654 | -1.0203195 |
Questions
What is the derivative of tan(-x)?
The derivative of is .
What is the second derivative of tan(-x)?
2cos(-x) * sin(-x) / (cos(-x)^2)^2.
Which rule do you use for tan(-x)?
The chain rule.
What is the slope of tan(-x) at x = 1?
-3.4255188.