Find the derivative of the inverse of the given function at the given point? f(x)=5x+#3x^2#+7x+2 at (13,1) help me please

1 Answer
bp
Dec 25, 2017

#1/(2sqrt 69)#

Explanation:

Write #f(x)=y= 3x^2 +12x+2= 3(x^2 +4x)+2=3(x^2 +4x +4) -10=3(x+2)^2 -10#

Thus, the given function is #y=3(x+2)^2 -10#. For finding inverse of f(x), interchange 'x' and 'y' ad the solve for 'y'. Accordingly, it would be #x=3(y+2)^2 -10#. Therefore #f^-1(x)#=#y=sqrt((x+10)/3) -2#. This represents an horizontal parabola with its vertex at (-10,-2).

derivative would be #y'=1/sqrt 3 1/2 (x+10)^(-1/2)#

At the given point (13,1) #y'=1/sqrt 3 1/2 1/sqrt 23=1/(2sqrt 69)#

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