## 3.1 Differentiation Formulae

The following rules allow us to find algebraic formulae for the derivative of most differentiable functions we know how to write down.

### 3.1.1 Derivative of Constant Function

, for any constant c  Proof of 1

### 3.1.3 The Sum Rule

Proof of sum rule

Example of sum rule

### 3.1.4 The Product Rule

In particular

Proof of product rule

Example of product rule

### 3.1.5 The Chain Rule

y = f(u), u = g(x), f and g differentiable.

Then

Example of chain rule

Proof of chain rule

### 3.1.6 Implicit Differentiation

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

or

Comment on implicit differentiation

Examples of implicit differentiation

### 3.1.7 The Quotient Rule

In particular,

Proof of quotient rule

Example of quotient rule

### 3.1.8 The Power Rule

for any power n, integer, rational or irrational.

hence,

implies

Proof of power rule

Example of power rule