| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2025 |
| Session | October |
| Marks | 4 |
| Topic | Differentiation from First Principles |
| Type | First principles: polynomial with multiple terms |
| Difficulty | Moderate -0.5 This is a straightforward application of the definition of derivative using first principles. While it requires careful algebraic manipulation of the difference quotient and expanding (x+h)³, it's a standard technique taught early in calculus with no conceptual difficulty—just methodical execution of a learned procedure for a polynomial function. |
| Spec | 1.07g Differentiation from first principles: for small positive integer powers of x |
Given the function $f(x) = 3x^3 - 7x - 1$, defined for all real values of $x$, prove from first principles that $f'(x) = 9x^2 - 7$. [4]
\hfill \mbox{\textit{SPS SPS FM 2025 Q3 [4]}}