Differentiation


   
 
Derivative of Some Important Functions
Derivative of a Constant
 
Let f(x) = k be the given function.
 
Then, we have
 
 
 
Therefore derivative of a constant is 0.
 
or
 
 
Derivative of xn where n is any integer
 
 
 
 
 
 
 
 
Note:
 
(i) The result is also true for any real exponent n.
 
(ii) Derivative of xn, where n is any real number, can be obtained by subtracting 1 from the exponent n and multiply the result by n.
 
Example:
 
 
Derivative of a Constant of a Function
 
Let f (x) = k u(x)
 
where k is a constant and u = u(x), xR, be a differential function of x. We then have
 
 
 
 
Derivative of Exponential Function
 
If f(x) = ex, then
 
 
 
 
 
 
Derivative of the Logarithmic Function
 
Let f(x) = loge x (x > 0). Then
 
 
 
 
 
 
Note:
 
Derivative of loga x (x > 0).
 
 
 
Derivative of sin x
 
Let f(x) = sin x. Then, we have
 
 
 
 
 
Remark: Similarly, one can prove that
 
 
Derivative of tan x
 
Let f(x) = tan x. Then, we have
 
 
 
 
 
 
 
 
Remark: Similarly, we can prove that
 
 
Derivative of sec x
 
Let f(x) = sec x. Then, we have
 
 
 
 
 
 
 
 
 
Similarly, we can prove that
 
 
Derivative of Sum of Two Functions
 
Let y = u + v, where u and v are differentiable functions of x.
 
 
 
 
 
 
or (u + v)' = u' + v'
 
Note:
 
In general, the result is true for the sum of any number of functions.
 
Example:
 
Differentiate the following function with respect to x.
 
 
Suggested answer:
 
 
 
 
 
  
     
   
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