Question: Answer all parts of this question Let \(F(x) \equiv \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} e^{-\frac{1}{2} y^{2}} d y\), where \(e\) is the exponential function, \(\exp (\cdot)\).

1. Use the Fundamental Theorem of Calculus to derive an expression for \(f(x) \equiv\) \(F^{\prime}(x)\).
2. How are \(F(x)\) and \(f(x)\) usually referred to?
   Let \(M(s, t)\) be a function of \(s\) and \(t\), defined as follows:
   \[M(s, t) \equiv \frac{\ln \frac{s}{k}+\left(r+\frac{1}{2} \sigma^{2}\right)(T-t)}{\sigma \sqrt{T-t}}\]
   where k, r, T and \(\sigma\) are constants.
3. Derive the partial derivatives \(\frac{\partial M}{\partial s}\) and \(\frac{\partial N}{\partial s}\), where \(N(s, t) \equiv M(s, t)-\sigma \sqrt{T-t}\).
   Finally, assemble the above functions into:
   \[G(s, t) \equiv s \cdot F(M(s, t))-k \cdot e^{-r(T-t)} F(N(s, t))\]
4. Derive the partial derivative \(\frac{\partial G}{\partial s}\). Hint: begin by applying the rules of differentiation to equation 2 to obtain an expression in \(f(), \frac{\partial M}{\partial s}\), and so on; you should obtain three terms; the third term simplifies considerably if you can substitute out \(N\), and then \(M\).

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