The following rules allow us to find algebraic formulae for the derivative of most differentiable functions we know how to write down.
, for any constant c Proof of 1
Proof of 2
Proof of sum rule
Example of sum rule
Proof of product rule
Example of product rule
y = f(u), u = g(x), f and g differentiable.
Example of chain rule
Proof of chain rule
Suppose the function f(x) is defined by an equation: g(f(x),x)=0, rather than by an explicit formula.
Then g is a function of two variables, x and f.
Thus g may change if f changes and x does not, or if x changes and f does not.
Let the change in g arising from a change, df, in f and none in x be a(f,x)df, and let the change in g from a change, dx, in x and none in f be b(f,x).
The total change in g must vanish since g is a constant, (0), which gives us
a(f,x)df + b (f,x)dx = 0
Comment on implicit differentiation
Examples of implicit differentiation
Proof of quotient rule
Example of quotient rule
for any power n, integer, rational or irrational.
Proof of power rule
Example of power rule