Question

Find the nth derivative of the following : a^px+q 

Answer 1

Let y = a^px+q 

Then `"dy"/"dx" = "d"/"dx"(a^(px + q))`

= `a^(px + q)loga."d"/"dx"(px + q)`

`(d^2y)/(dx^2) = "d"/"dx"[ploga.a^(px + q)]`

= `ploga."d"/"dx"(a^(px + q))`

= `ploga.a^(px + q).log a."d"/"dx"(px + q)`

= `ploga.a^(px + q).log a xx (p xx 1 + 0)`

= `p^2.(loga)^2.a^(px + q)`

`(d^3y)/(dx^3) = "d"/"dx"[p^2.(loga)^2.a^(px + q)]`

= `p^2.(loga)^2."d"/"dx"(a^(px + q))`

= `p^2.(log a)^2.a^(px + q).log a."d"/"dx"(px + q)`

= `p^2.(loga)^3.a^(px + q) xx (p xx 1 + 0)`

= `p^3.(loga)^3.a^(px + q)`
In general, the nth order derivative is given by
`(d^ny)/(dx^n) = p^n.(loga)^n.a^(px + q)`.