Differentiation

• if p = f(T) then
  
  $$\frac{d p}{d T} = \frac{d f(T)}{d T} = f'(T) = \lim_{h \rightarrow 0} \frac{f(T+h)- f(T)}{h}$$
  
  

• if h(T) and g(T) are functions of T, then
  
  $$\frac{d (h(T) \times g(T))}{d T} = \frac{d h(T)}{d T} g(T) + h(T) \frac{d g(T)}{dT}$$
  
  

• if i(T) and j(T) are functions of T, then
  
  $$\frac{d (i(T)/j(T) )}{d T} = \frac{1}{j(T)}\frac{d i(T)}{d T} - \frac{i(T)}{j(T)^2} \frac{d j(T)}{d T}$$
  
  

• if $k = \ell(T)$, and T is a function of y, i.e., T = T(y), then ¹
  
  $$\frac{d k}{d y} = \frac{d \ell(T)}{dy} = \frac{d \ell(T)}{dT}\frac{dT(y)}{d y}$$
  
  

• if y = f(x) has a solution x = g(y) then
  
  $$\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}$$
  
  

• Here is a summary of the relationships between functions and their first and their second derivatives:
  
   

  Figure 1: Shapes of Curves and Their Derivative Values(I)

  
  
  
  
   

  Figure 2: Shapes of Curves and Their Derivative Values(II)

  
  
  
• at extrema, the derivative of a differentiable function is zero. For relative (local) maxima, the second derivative is less than zero, while for relative minima, the second derivative is greater than zero.