Question

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

Answer 1

`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)`