Consider the Black-Scholes call pricing formula

$$
C(T-t, x, K)=K f\left(T-t, \frac{x}{K}ight)
$$

written using the function

$$
f(\tau, z):=z \Phi\left(\frac{\left(r+\sigma^{2} / 2ight) \tau+\log z}{|\sigma| \sqrt{\tau}}ight)-\mathrm{e}^{-r \tau} \Phi\left(\frac{\left(r-\sigma^{2} / 2ight) \tau+\log z}{|\sigma| \sqrt{\tau}}ight) .
$$

a) Compute $\frac{\partial C}{\partial x}$ and $\frac{\partial C}{\partial K}$ using the function $f$, and find the relation between $\frac{\partial C}{\partial K}(T-$ $t, x, K)$ and $\frac{\partial C}{\partial x}(T-t, x, K)$.

b) Compute $\frac{\partial^{2} C}{\partial x^{2}}$ and $\frac{\partial^{2} C}{\partial K^{2}}$ using the function $f$, and find the relation between $\frac{\partial C^{2}}{\partial K^{2}}(T-$ $t, x, K)$ and $\frac{\partial C^{2}}{\partial x^{2}}(T-t, x, K)$.

c) From the Black-Scholes PDE

$$
\begin{aligned}
r C(T-t, x, K)= & \frac{\partial C}{\partial t}(T-t, x, K)+r x \frac{\partial C}{\partial x}(T-t, x, K) \\
& +\frac{\sigma^{2} x^{2}}{2} \frac{\partial^{2} C}{\partial x^{2}}(T-t, x, K),
\end{aligned}
$$

recover the Dupire (1994) PDE for the constant volatility $\sigma$.