Differentiation Rules - Practice Questions & MCQ

Rules of Differentiation (Sum/Difference/Product)

Let f(x) and g(x) be differentiable functions and k be a constant. Then each of the following rules hold:

Sum Rule

The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.

In general,

Difference Rule

The derivative of the difference of a function f and a function g is the same as the difference of the derivative of f and the derivative of g.

Constant Multiple Rule

The derivative of a constant k multiplied by a function f is the same as the constant multiplied by the derivative of f

Product rule

Let f(x) and g(x) be differentiable functions. Then,

This means that the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function

Extending the Product Rule

If 3 functions are involved, i.e let k(x) = f(x)・ g(x)・h(x) 

Let us have a function k(x) as the product of the function f(x)g(x) and the function h(x). That is, k(x) = (f(x)・ g(x))・h(x).  Thus, 

Rules of Differentiation (Division or Quotient Rule)

Quotient Rule

Let f(x) and g(x) be differentiable functions. Then

OR

As we see in the following theorem, the derivative of the quotient is not the quotient of the derivatives.

If   and  are differentiable funcitons, then  or   is also differentiable.

If  , then 

is known as chain rule.  Or,

If   and    then  

The chain rule can be extended as follows:

If   then