Derivatives
(cod: P-52-29-11)
Consider the following questions:
(1) \(\dfrac{d}{dx}\left(\dfrac{2x^2}{x^2 - 9}\right)\) is equal to:
(2) \(\dfrac{d}{dx}\left(e^{-2x}\cos(3x)\right)\) is equal to:
(3) Let \(F\), \(G\), and \(H\) be differentiable functions such that:
\begin{eqnarray*}
F(0) &=& 2, \ \ \ \ \ \ \ F'(0) &=& 3,
\\ G(0) &=& 1, \ \ \ \ \ \ \ G'(0) &=& -1,
\\ G(1) &=& -1, \ \ G'(1) &=& 1,
\\ H(0) &=& 1, \ \ \ \ \ \ \ H'(0) &=& 2.
\end{eqnarray*}
If \(f(x) = F(x)G(H(x))\), determine \(f'(0)\).
(4) Let \(f : I \to J\) be a differentiable bijection between open intervals \(I\) and \(J\) such that
\[f(1) = 0 \ \ \ \text{and} \ \ \ f'(1) = \dfrac{1}{2}.\] Denote by
\(f^{-1}:J \to I\) the inverse of \(f\) and consider the function \(g(t) = (t^2 + t + 3)f^{-1}(t)\). Determine the value of \(g'(0)\).
Select the alternative that correctly answers the four questions above.