How do you differentiate sin(x) - ln(2x)

Question

Please explain how to do this sir

Answer 1

The function  f(x) =  sin(x) - ln(2x)   is the difference of two functions (namely sin(x) and ln(2x) )

 

The derivative of a difference  is the difference of the derivatives.   The derivative of sin(x) is  cos(x)

 

The derivative of the natural log  ln(x) is   1/x.   There are two ways to get the derivative of ln(2x) from this.  

   The east way is to note that ln(2x) =  ln(2) + ln(x).   Since the derivative of the constant ln(2) is zero, we find that the derivative of  ln(2x) is the same as that of ln(x) , namely 1/x.

 

  Putting this together, we get that the derivative of f(x) is  cos(x) - 1/x

 

The other way to work out the derivative of ln(2x) is the chain rule.   According to the chain rule

the derivative of ln(2x) is    1/(2x) times the derivative of 2x (namely 2).   Thus the derivative of

ln(2x) is   [1/(2x)] 2 =   1/x   in agreement with the easy way above.