What is the slope of the line tangent to the curve y = arctan(4x) where x = 1/4?

The slope of the line tangent to the curve y = arctan(4x) where x = 1/4 is 2

Given the function

The slope of the function can be given by differentiating the function as shown below;

On differentiating the given function

Substitute x = 1/4 into the result to have:

Hence the slope of the line tangent to the curve y = arctan(4x) where x = 1/4 is 2

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solution:

the given function is ,

      =

the three roots of this quadratic function is , 0 ,8 and 10.all the roots are real and rational.

if a quadratic equation is of the form ,

sum of the roots = a + b +c= 0+8+10=18=  

product of roots taken two at a time = a b + b c+ c a = 0+80+0=80 =  

product of roots = a b c = 0=  

there are 11 doors in a hall starting from 0 to 11.the entry in the hall is permitted through gate number 0 and exit is allowed through only gate number 8 and 10.so, three roots(0,8,10) of quadratic function is given by entry and exit route of this hall.