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
\[ \frac{d}{dx}\left[\tan\left(5 x\right)\right] = \frac{5}{\cos\left(5 x\right)^{2}} \]
\[ f(x) = \tan\left(5 x\right) \]
\[ g(x) = 5 x,\quad g'(x) = 5 \]
\[ f'(x) = \left[\frac{1}{\cos\left(u\right)^{2}}\right]_{u=g(x)} \cdot g'(x) \]
\[ f'(x) = \frac{5}{\cos\left(5 x\right)^{2}} \]
\[ f''(x) = \frac{-\left(5 2 \cos\left(5 x\right) -\left(5 \sin\left(5 x\right)\right)\right)}{\left(\cos\left(5 x\right)^{2}\right)^{2}} \]Values of f and f' for tan(5x)
| x | f(x) | f'(x) |
|---|---|---|
| -2 | -0.64836083 | 7.1018588 |
| -1 | 3.380515 | 62.139409 |
| 0.5 | -0.7470223 | 7.7902116 |
| 1 | -3.380515 | 62.139409 |
| 2 | 0.64836083 | 7.1018588 |
| 3 | -0.8559934 | 8.6636235 |
Questions
What is the derivative of tan(5x)?
The derivative of is .
What is the second derivative of tan(5x)?
-(5 * 2cos(5x) * -(5sin(5x))) / (cos(5x)^2)^2.
Which rule do you use for tan(5x)?
The chain rule.
What is the slope of tan(5x) at x = 1?
62.139409.