Using the limit definition, how do you differentiate #f(x)=sqrt(x+1)#?

2 Answers
May 27, 2016

#lim_{Delta->0}((sqrt(x+1+Delta)-sqrt(x+1)))/Delta=1/(2sqrt(x+1))#

Explanation:

Let us calculate the limit indirectly.
First we calculate the limit:

#l(x)=lim_{Delta->0}(sqrt(x+1+Delta)+sqrt(x+1))times lim_{Delta->0}((sqrt(x+1+Delta)-sqrt(x+1)))/Delta#
This limit is equivalent to
#lim_{Delta->0}((x+1+Delta)-(x+1))/Delta=1#
but
#lim_{Delta->0}(sqrt(x+1+Delta)+sqrt(x+1)) = 2sqrt(x+1)#
Putting all together
#l(x)=2sqrt(x+1) lim_{Delta->0}((sqrt(x+1+Delta)-sqrt(x+1)))/Delta = 1#
and finally
#lim_{Delta->0}((sqrt(x+1+Delta)-sqrt(x+1)))/Delta=1/(2sqrt(x+1))#

May 27, 2016

See explanantion

Explanation:

To solve this you use 2 general principles.

#color(blue)("Principle 1:")#

Multiply any number by 1 and you do not change its value. However you can change the way it looks. Consider a basic case of #3xx1=3#

Multiply 3 by 1 but in the form of #1=sqrt(2)/sqrt(2) -> 3xxsqrt(2)/sqrt(2) =(3sqrt(2))/sqrt(2) #

'.......................................................................
#color(blue)("Principle 2:")#

Suppose you had #(a+b)#

Multiply it by 1 but in the form iof #1=(a-b)/(a-b)#

Then #(a+b)(a-b)/(a-b) = (a^2-b^2)/(a-b)#
#color(red)("Your question has square roots and to get rid of this you square it!")#
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given:#" "f(x)=sqrt(x+1)#

Write as #y=sqrt(x+1)# ..................................(1)

Increment your values giving

#y+delta y =sqrt(x+delta x+1)#.........................(2)

Equation (2) - Equation (1)

#delta y=sqrt(x+delta x + 1) - sqrt(x+1)#

Divide throughout by #delta x#

#(delta y)/(delta x) = (sqrt(x+delta x + 1) - sqrt(x+1))/(delta x)# .......(3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Multiply equation (3) by 1 but where #1=(sqrt(x+delta x + 1) + sqrt(x+1))/(sqrt(x+delta x + 1) + sqrt(x+1))#

Now we have our #(a-b)(a+b)=a^2-b^2# condition giving:

#(delta y)/(delta x) = ((x+delta x + 1) - (x+1))/(deltax(sqrt(x+delta x + 1) + sqrt(x+1)))#

#(delta y)/(delta x) =( cancel(x)+cancel(delta x) +cancel( 1) -cancel( x)-cancel(1))/(cancel(deltax)(sqrt(x+delta x + 1) + sqrt(x+1)))#

#lim_(delta x->0) (deltay)/(deltax)= 1/(sqrt(x+1)+sqrt(x+1)) = 1/(2sqrt(x+1))#

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Which is of form # d/(dx) (x^n) = nx^(n-1)#