| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | April |
| Marks | 11 |
| Topic | Differentiating Transcendental Functions |
| Type | Differentiate exponential functions |
| Difficulty | Moderate -0.3 This is a standard calculus techniques question testing routine differentiation (chain rule, product rule, quotient rule) and integration (substitution, polynomial division). All parts follow textbook patterns with no problem-solving required, but the variety of techniques and algebraic manipulation needed (especially parts i(d) and ii(b)) elevate it slightly above pure recall, placing it just below average difficulty. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07l Derivative of ln(x): and related functions1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08j Integration using partial fractions |
\begin{enumerate}[label=(\roman*)]
\item Differentiate the following with respect to $x$, simplifying your answers fully
\begin{enumerate}[label=(\alph*)]
\item $y = e^{3x} + \ln 2x$ [1]
\item $y = (5 + x^2)^{\frac{3}{2}}$ [1]
\item $y = \frac{2x}{(5-3x^2)^{\frac{1}{2}}}$ [2]
\item $y = e^{-\frac{3}{x}} \ln(1 + x^3)$ [2]
\end{enumerate}
\item Integrate with respect to $x$
\begin{enumerate}[label=(\alph*)]
\item $\frac{7}{(2x-5)^8} - \frac{3}{2x-5}$ [2]
\item $\frac{4x^2+5x-3}{2x-5}$ [3]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2021 Q1 [11]}}