Question

If f(x) = 2|x| + 3|sin x| + 6, then the right hand derivative of f(x) at x = 0 is ______.

Answer 1

If f(x) = 2|x| + 3|sin x| + 6, then the right hand derivative of f(x) at x = 0 is 5.

Explanation:

f(x) = 2|x| + 3|sin x| + 6

f(x) = 2|0| + 3|sin 0| + 6

= 0 + 3(0) + 6

= 0 + 0 + 6 

= 6

R.H.D. = `lim_(h -> 0) (f(0 + h) - f(0))/h`

= `lim_(h -> 0) ([2|0 + h| + 3|sin(0 + h)| + 6] - [6])/h`

= `lim_(h -> 0) (2h + 3 sin h + 6 - 6)/h`

= `lim_(h -> 0) (2h)/h + lim_(h -> 0) (3 sin h)/h`

= `2 + 3 lim_(h -> 0) (sin h)/h`   ...`[lim_(x -> 0) sin x/x = 1]`

= 2 + 3(1)

= 2 + 3

= 5