Question

Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = 3x² + 2logx³ 

Answer 1

y = 3x² + 2logx³ 
= 3x² + 6logx
Differentiating w.r.t. x, we get
`"dy"/"dx" = "d"/"dx"(3x^2 + 6logx)`

= `3 xx 2x + 6 xx 1/x`

= `6x + 6/x`

= `(6x^2 + 6)/x`
The derivative of inverse function of y = f(x) is given by
`"dx"/"dy" = (1)/(("dy"/"dx")`

= `(1)/(((6x^2 + 6)/x)`

= `x/(6x^2 + 6)`

At x = `1, "dx"/"dy"`

= `(x/(6x^2 + 6))_(at  x = 1)`

= `(1)/(6(1)^2 + 6)`

= `(1)/(12)`.