Consider the functional equation #f(x-y)=f(x)/f(y)# . If #f'(0) = p# and #f'(5)=q# , then find #f'(-5)# ?

2 Answers
Cesareo R.
May 2, 2017

#f'(-5)=p^2/q#

Explanation:

Making #x=y# we have

#f(0)=1#

Making #x=2y# we have

#f(y)=f(2y)/f(y)# or

#f(2y)=f(y)^2#

or

#f(ky)=f(y)^k# for #k in ZZ#

but also with #x=0#

#f(-y)=f(y)^-1# and

#f(-ky)=f(y)^-k#

Supposing now

#f(y) = a^y#

we have

#d/(dy)f(y)=a^ylog_e a->a^0log_ea=p#

or

#a = e^p#

and

#f(y) = (e^p)^y#

then

#f'(5)=(e^p)^5 p=q# and

#f'(-5)=p/(e^p)^5#

or

#f'(-5) = p^2/q#

Ratnaker Mehta
May 2, 2017

# p^2/q.#

Explanation:

Given that, #f(x-y)=f(x)/f(y).......(star).#

#x=y, &, (star) rArr f(0)=f(y)/f(y)=1.......(1).#

#x=0, y=y, & (star) rArr f(-y)=f(0)/f(y)=1/f(y).........(2).#

#x=x, y=-y, & (star) rArr f(x+y)=f(x)/f(-y)=f(x)f(y)#

#:.," by "(2), f(x+y)=f(x)f(y).............(3).#

Knowing that, #f'(x)=lim_(h to 0) {f(x+h)-f(x)}/h,# we have, by #(3),#

#f'(x)=lim_(h to 0){f(x)f(h)-f(x)}/h=f(x){lim_(h to 0) (f(h)-1)/h}...(ast).#

Now, #f'(0)=p, (ast), & (1) rArr p=lim_(h to 0) (f(h)-1)/h...(4).#

Similarly, #f'(5)=q, (ast), & (4) rArr q=p*f(5)......(5).#

#"Finally, by "(ast), &, (4), f'(-5)=p*f(-5)#

#=p*(1/f(5)),............[because,(2)]#

#=p/(q/p),................[because, (5)]#

#:. f'(-5)=p^2/q,# as Respected Cesareo R., has derived!

Enjoy Maths.!

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