How to find f'(0) ?

Answer 1

#f'(0)=-6#

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

and using quotient rule,

#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#
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Answer 2

#f'(0) = -6#

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

Using the quotient rule:

#(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#
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Answer 3

#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.!

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Answer 4

To find ( f'(0) ), you need to differentiate the function ( f(x) ) with respect to ( x ) and then evaluate it at ( x = 0 ). So, if you have the function ( f(x) ), differentiate it to get ( f'(x) ), and then substitute ( x = 0 ) into the derivative function to find ( f'(0) ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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