To determine

To evaluate:

a) The derivative f'(x)

b) Compare the graphs of functions f and f'

Answer

The value of the derivative of f is f'(x)=cosx+1x

Formula used:

1. Addition rule:u+v'=u'+v'

Explanation

Given:

f(x)=sinx+lnx

Calculation:

First, we find the derivative f'(x). Consider

f(x)=sinx+lnx

Differentiate it with respect to x, we get

f'(x)=ddx[sinx+lnx]=ddsinx+ddxlnx                                               (By using addition rule)=cosx+1x

Consider,

We know that derivative denotes the slope of the tangent line at that point. In the above graph also, we can see this. For example in the above graph there are three visible points where the tangent lines of f is parallel to the x-axis i.e. slope of the tangent lines at that points is zero and at that points value of f'(x)=cosx+1x is zero. Also, when f' is positive, f is increasing and when f' is negative, f is decreasing. This clearly, shows that f'(x)=cosx+1x is derivative of f(x)=sinx+lnx.

Conclusion: The value of the derivative of f is f'(x)=cosx+1x