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Question
Find the second order derivatives of the following : x3.logx
Solution
Let y = x3.logx
Then, `"dy"/"dx" = "d"/"dx"(x^3.logx)`
= `x^3"d"/"dx"(logx) + (logx)."d"/"dx"(x^3)`
= `x^3 xx (1)/x + (logx) xx 3x^2`
= x2 + 3x2 log x
= x2(1 + 3 log x)
and
`(d^2y)/(dx^2) = "d"/"dx"[x^2(1 + 3logx)]`
= `x^2."d"/"dx"(1 + 3logx) + (1 + 3logx) xx 2x`
= `x^2(0 + 3 xx 1/x) + (1 + 3logx) xx 2x`
= 3x + 2x + 6x log x
= 5x + 6x log x
= x(5 + 6 log x).
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