Question

Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = x⁵ + 2x³ + 3x, at x = 1

Answer 1

y = x⁵ + 2x³ + 3x
Differentiating w.r.t. x, we get
`"dy"/"dx" ="d"/"dx"(x^5 + 2x^3 + 3x)`
= 5x⁴ + 2 x 3x² + 3 x 1
= 5x⁴ + 6x² + 3
The derivative of inverse function of y = f(x) is given by
`"dx"/"dy" = (1)/(("dy"/"dx")`

= `(1)/(5x^4 + 6x^2 + 3)`

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

= `(1)/(5x^4 + 6x^2 + 3)_(at  x = 1)`

= `(1)/(5(1)^4 + 6(1)^2 + 3)`

= `(1)/(5 + 6 + 3)`

= `(1)/(14)`.