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Question
Find the nth derivative of the following : apx+q
Solution
Let y = apx+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)`.
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