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Can one use the Greeks (delta,gamma,theta) to show that the Black-Scholes call formula satisfies the Black-Scholes PDE?

If so, is there a derivation anywhere that shows this? I was told that this could be done in a class but I don't see how it's possible.
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Option greeks: sensitivity to 1% move

In a Black&Scholes framework how can I compute the following sensitivities: to 1% move in the underlying price to 1% move in implied volatility I would like the greeks to tell me how many ...