Some Simple Derivatives

• Definition: The rate of change [slope] of a function at a point is the limiting value of its average slope over an interval including that point, as the width of the interval shrinks to zero:
  
  $${dy \over dx} \; \equiv \; \lim_{\Delta x \to 0} \; {y(x + \Delta x) \; - \; y(x) \over \Delta x }$$
  
  
  All the remaining Laws and Rules can be proven by algebraic manipulation of this definition.

• Power Law:
  
  $${d \over dx} (x^p) \; = \; p \, x^{p-1}$$
  
  
  valid for all powers p, whether positive, negative, integer, rational, irrational, real, imaginary or complex.

• Product Law: The derivative of the product of two functions is not the product of their derivatives! Instead,
  
  $${d \over dx} \left[ f(x) \cdot g(x) \right] \; = \; {df \over dx} \cdot g(x) \; + \; f(x) \cdot {dg \over dx}$$
  
  

• Constant times a Function: The Product Law gives
  
  $${d \over dx} \left[ a \, y(x) \right] \; = \; a \, {dy \over dx}$$
  
  
  where a is a constant (, not a function of x). This is often referred to as "pulling the constant factor outside the derivative."

 

• Function of a Function: Suppose y is a function of x and x is in turn a function of t.
  
  $${d \over dt} \, y[x(t)] \; = \; {dy \over dx} \cdot {dx \over dt}$$
  
  
  (also known as the Chain Rule)

• Exponentials:
  
  $${d \over dx} \left[ e^{k \, x} \right] \; = \; k \, e^{k \, x}$$
  
  
  where k is any constant.

• Natural Logarithms:
  
  $${d \over dx} \left[ \ln x \right] \; = \; {1 \over x}$$