Higher Order derivatives

Calculus Level 2

What is the number of times the function f ( x ) = ( x 2 + 1 ) 80 f(x) = (x^2+1)^{80} to be differentiated to result in polynomial of degree 50?

80 50 30 110 70

Rubal Singh Dhaliwal by Rubal Singh Dhaliwal , 20, India

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1 solution

f n ( x ) = 80 × 79 × 78 × . . ( 80 − n ) × 2 n ( x n ( x 2 + 1 ) 80 − n ) ⟹ x n ( x 2 ( 80 − n ) ) = x 50 ⟹ x 160 − n = x 50 ⟹ n = 110 f^{n}(x) = 80 \times 79 \times 78 \times ..(80-n) \times 2^{n} (x^{n}(x^{2}+1)^{80-n}) \Longrightarrow x^{n}( x^{2(80-n)}) = x^{50} \Longrightarrow x^{160-n} = x^{50} \Longrightarrow n = 110

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