Q.232. Fourth derivative

Question 232.
Find the 4th derivative of (6x^4)/(1-x) without using the quotient rule 4 times.

Answer 232.
y
= 6x^4 / (1 - x)
= - 6 (1 - x^4 - 1) / (1 - x)
= - 6[(1 - x)(1 + x + x^2 + x^3) - 1] / (1 - x)
= - 6(1 + x + x^2 + x^3) + 6/(1 - x)
Fourth derivative of the first term in the bracket = 0
=> d^4y/dx^4
= 4th derivative of 6/(1 - x)
First derivative = 6/(1 - x)^2
Second derivative = 12/(1 - x)^3
Third derivative = 36/(1 - x)^4
Fourth derivative = 144/(1 - x)^5.

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