The standard derivatives are listed below. For trigonometric functions, these results are valid only when is measured in radians, as required in IB Mathematics: Applications and Interpretation HL calculus.
| Function | Derivative (f'(x)) | Important condition |
|---|---|---|
| is measured in radians |
| Keep the negative sign |
| Defined where |
| Its own derivative |
| Requires |
| may be rational |
These are the standard derivatives in AHL 5.9. A derivative gives a function’s instantaneous rate of change, equivalently the gradient of its tangent. Differentiation is linear, so sums and constant multiples can be differentiated term by term.
For example, let . Applying the rules separately gives . The coefficient remains unchanged, and the power rule changes to .
When the input is not simply , apply the chain rule by multiplying by the derivative of the inner function. For example, when , because has derivative . Similarly, when .
The general chain-rule patterns include , , and .
These formulas can also be combined with the product rule and quotient rule. If , then . Both factors must be differentiated in separate terms; differentiating only one factor is insufficient.
Always consider the domain. Although , the original real-valued function requires . Likewise, and are undefined where . These restrictions matter when solving derivative equations or interpreting gradients.
Two common misconceptions are differentiating as and omitting the inner derivative. The correct first derivative is . Another error is applying the power rule to ; its derivative remains .
Exam technique: Show the derivative rule and each required multiplier, then simplify. A GDC is permitted on every Mathematics AI paper, but calculator output does not replace mathematical working. For trigonometric differentiation, check radian mode before evaluating derivatives.
