How to find f'(0) ?

Question

38601 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-03-18T15:05:47

#f'(0)=-6#

As #f(x)=(x^2+3)/(2x-1)#

#f'(x)=(2x xx(2x-1)-2xx(x^2+3))/(2x-1)^2#

= #(4x^2-2x-2x^2-6)/(2x-1)^2#

= #(2x^2-2x-6)/(2x-1)^2#

= #(2(x^2-x-3))/(2x-1)^2#

and #f'(0)=(2(0^2-0-3))/(2xx0-1)^2=-6/1=-6#

Answer 2 · 2017-03-18T15:11:17

#f'(0) = -6#

#f(x) = (x^2+3)/(2x-1)#

#(df)/dx = ((2x-1) d/dx(x^2+3) - (x^2+3) d/dx(2x-1))/(2x-1)^2#

#(df)/dx = (2x (2x-1) - 2(x^2+3))/(2x-1)^2#

#(df)/dx = (4x^2 -2x - 2x^2-6)/(2x-1)^2#

#(df)/dx = (2x^2 -2x -6)/(2x-1)^2#

and for #x=0#

#[(df)/dx]_(x=0) = -6#

Answer 3 · 2017-03-18T15:21:56

#f'(0)=-6.#

Recall that, #f'(a)=lim_(x to a) {(f(x)-f(a))/(x-a)}.#

#:. f'(0)=lim_(xto0){(f(x)-f(0))/(x-0)}=lim_(xto0){(f(x)-f(0))/x}.#

# f(x)=(x^2+3)/(2x-1) rArr f(0)=(0+3)/(0-1)=-3.#

#:. f'(0)=lim_(xto0)[1/x{(x^2+3)/(2x-1)-(-3)}],#

#=lim_(xto0)1/x[{x^2+3+3(2x-1)}/(2x-1)],#

#=lim_(xto0)1/x{(x^2+6x)/(2x-1)},#

#=lim_(xto0){x(x+6)}/{x(2x-1)},#

#=lim_(xto0){(x+6)/(2x-1)}.#

# rArr f'(0)=(0+6)/(0-1)=-6.#

Enjoy Maths.!