Grade 12th passDifferential Calculus
n th derivative of y = xlog(x+1)Prove that yn = (-1)n (n-2)! (x+n)÷(x+1)n
n th derivative of y = xlog(x+1)
Prove that
yn = (-1)n (n-2)! (x+n)÷(x+1)n
1 Answer
Himanshu Gharat
y=xlog(x+1)
y1=x(x+1)-1 +log(x+1)
y2=(x+1-x)(x+1)-2+(x+1)-1
y2=(x+1)-2 + (x+1)-1
we know that
(ax+b)-m=((-1)n an (m+n-1)!) × ((ax+b)m+n (m-1)!)-1
Differentiating w.r.t. x (n-2) times
yn=((-1)n-2 (2+n-2-1)!) × ((x+1)2+n-2(2-1)!)-1 + ((-1)n-2 (n-2)!) × ((x+1)n-2+1)-1
yn=((-1)n (n-1)! ) × (x+1)-n + ((-1)n (n-2)! × ((x+1)n-1)-1)
yn=((-1)n (n-1) (n-2) ! ×(x+1)-n) + ((-1)n (n-2)! (x+1) × (x+1)-n )
yn = {(-1)n (n-2)! × (x+1)-n } (n-1+x+1)
yn = { (-1)n (n-2)! (n+x) } (x+1)-n
Hence prooved
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