Rules of Differentiation

In order to simplify the task of finding derivatives, some general rules are developed to help finding derivatives with having to use the defination directly.

Derivatives of Polynomials

1. Derivative of Constant Function

   

   Proof:

   

2. Derivative of Linear Function

   

   Proof:

   

3. Derivative of Power Function

   

   Proof when n is positive:

   

   Proof when n is zero and x not equal to zero:

   

   n can be any real number and proof will be included in the later part of rules.

4. Rule of Constant Multiple Function

   

   Proof:

   

5. Rule of Sum of Functions

   

   Proof:

   

   The sum of functions can be extended to the sum of any functions by repeating the sum rule.

6. Rule of Difference of Functions

   

   Proof if both functions are differentiable:

   

   The difference of functions can be extended to the difference of any functions by repeating the difference rule. And the different rule can be obtained by applying the constant multiple rule to the sum rule. 

7. Rule of Product of Functions

   

   Proof if both functions are differentiable:

   

   The proof is completed by subtracting and adding an addition expression in the numerator. Since both functions f and g are differentiable, they are continuous and f(x)=f(x+Δx) and g(x)=g(x+Δx) when Δx approaching zero. The product rule can also be extend to any number of functions.

8. Rule of Quotient of Functions

   

   Proof if both functions are differentiable:

   

   The proof is completed by subtracting and adding an addition expression in the numerator. Since function g is differentiable, it is continuous and g(x)=g(x+Δx) when Δx approaching zero.

   Proof for the power function when n is negative:

   

9. Rule of Composite Function (The chain rule):

   

   Proof if both functions are differentiable:

   For g(x)

   

   For f(g(x)):

   

   Therefore:

   

   Since function g is differentiable, it is continuous. When Δx approaching zero, εg equals zero and Δh is approaching zero. And therefore εf equals zero as Δx approaching zero also. The rule of composite function can be extended to any number of composite function and the derivative can be expressed in form of a chain.

10. Rule of Inverse Function

    

    Proof if both functions are differentiable and x is the inverese function of y:

    

    Since x=f-1(y) if and only if y=f(x):

    

    Therefore:

    

    Proof for the power function when n is rational number: