Risk Neutral Pricing and Financial Mathematics: A Primer

Table 1.1

Sample Derivative Rules (c and n are Arbitrary Constrants)

1. $\frac{\mathrm{d}}{\mathrm{d}x}\left[x^n\right]=n{x}^{n-1}(\text{power rule})$ 2. $\frac{\mathrm{d}}{\mathrm{d}x}[f(x)+g(x)]=\frac{\mathrm{d}}{\mathrm{d}x}[f(x)]+\frac{\mathrm{d}}{\mathrm{d}x}[g(x)](\text{sum rule})$ 3. $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\mathrm{d}y}{\mathrm{d}u}\frac{\mathrm{d}u}{\mathrm{d}x}(\text{chain rule})$ 4. $\frac{\mathrm{d}}{\mathrm{d}x}[f(x)g(x)]=f(x)\frac{\mathrm{d}}{\mathrm{d}x}[g(x)]+g(x)\frac{\mathrm{d}}{\mathrm{d}x}[f(x)](\text{product rule})$ 5. $\frac{\mathrm{d}}{\mathrm{d}x}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)\frac{\mathrm{d}}{\mathrm{d}x}[f(x)]-f(x)\frac{\mathrm{d}}{\mathrm{d}x}[g(x)]}{{[g(x)]}^2}(\text{quotient rule})$ 6. $\frac{\mathrm{d}}{\mathrm{d}x}[cf(x)=c\frac{\mathrm{d}}{\mathrm{d}x}[f(x)](\text{constant multiple ...}$