Given g(x) = x^99 + 100x^15 + 1557x^12 + 11548x^5 + 123456, what is the 100th derivative of g(x)?

1 Answer
MattyMatty
Jun 18, 2018

#0#

Explanation:

#"Each time we derive a positive integer power of x with respect"#
#"to x, the exponent decreases by 1, until we have only a"#
#"constant left and if we derive a constant we get 0."#

#f(x) = a x^n " (n positive integer) "=> f'(x) = a*n*x^(n-1)#

#"If we derive f(x) k times we get"#

#f^(k) (x) = a n (n-1) ...(n-k+1) x^(n-k)#

#"if k = n we get "x^0 = 1" and "#

#f^(n)(x) = a * n! = " constant (not dependent on x anymore)"#

#"If we derive a constant we get 0 and if we derive 0, it remains"#
#"0 so"#
#"if k > n we get 0."#

#"And here k > n for all terms so we get"#

#0 + 0 + ... + 0 = 0#

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