Double prime?

viet

New member
Joined
Nov 19, 2005
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13
the problem said to fine f ' (2), f '' (2), and f ''' (2)
f(x) = (1-3x)/(1+3x)

i used the quotient rule to fine f '(x):
f '(x) = (1+3x)(1-3x)' - (1+3x)'(1-3x) / (1+3x)^2
f '(x) = (1+3x)(-3) - (3)(1+3x) / (1+3x)^2
f '(x) = (-3-9x) - (3-9x) / (1+3x)^2
f '(x) = -6 / (1+3x)^2

f '(2) = -6 / (1+3(2))^2

the next part is to find f ''(2) and f '''(2)
im not sure how to do find that can someone help me with this? thanks in advance
 
so use the quotient rule again and apply it to f '(x) = -6 / (1+3x)^2 ?
sofar i got to:
f ''(x) = ((1+3x)^2)(-6)' - ((1+3x)^2)'(-6) / ((1+3x)^2)^2
 
Hello, viet!

You have: f(x)=6(1+3x)2\displaystyle \,f'(x)\:=\:-6(1\,+\,3x)^{-2}

Then: f(x)=12(1+3x)3(3)  =  36(1+3x)3=36(1+3x)3\displaystyle \,f''(x)\:=\:12(1\,+\,3x)^{-3}(3)\;=\;36(1\,+\,3x)^{-3}\:=\:\frac{36}{(1\,+\,3x)^3}

Hence: f(2)=36(1+32)3=  36343\displaystyle \,f''(2)\:=\:\frac{36}{(1\,+\,3\cdot2)^3} \:= \;\frac{36}{343}


Since we have: f(x)=  36(1+3x)3\displaystyle \,f''(x)\:=\;36(1\,+\,3x)^{-3},
    \displaystyle \;\; then: f(x)  =  108(1+3x)43  =  324(1+3x)4\displaystyle \,f'''(x)\;=\;-108(1\,+\,3x)^{-4}\cdot3\;=\;\frac{-324}{(1\,+\,3x)^4}

Hence: \(\displaystyle \,f'''(2)\:=\:\frac{-324}{(1\,+\,3\cdot2)^4}\;=\;-\frac{324}{2401\)
 
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