Key Idea: This topic adds the HL derivatives: tan, sec, csc, cot, aˣ, logₐ x and the inverse trig functions. Then you combine them with the product, quotient and chain rules on Paper 1 and Paper 2.
| Function | Derivative |
|---|---|
| tan x | sec²x |
| sec x | sec x tan x |
| csc x | −csc x cot x |
| cot x | −csc²x |
| aˣ | aˣ ln a |
| logₐ x | 1/(x ln a) |
| arcsin x, arccos x | ±1/√(1 − x²) |
| arctan x | 1/(1 + x²) |
Combining the new derivatives
- Two functions multiplied: u′v + uv′ (product).
- One divided by another: (u′v − uv′)/v² (quotient).
- One inside another: chain rule, outer′ × inside′.
The 'co-' functions csc and cot pick up a minus sign.
Gradient of y = arctan x at x = 1? 1/2.
Slope of y = tan x at x = π/4? sec²(π/4) = 2.
Derivative of (arctan x)/x? (x/(1 + x²) − arctan x)/x².
Why is d/dx(aˣ) not the power rule? x is in the exponent, so ln a appears.