Differentiate 100 times?

Calculus Level 1

What is the 10 0 t h 100^{th} derivative of f ( x ) = x 2019 f(x)=x^{2019} with respect to x x ?

2019 ! x 100 2019!x^{100} 2019 ! 100 ! x 1919 \frac{2019!}{100!}x^{1919} 100 ! x 1919 100!x^{1919} 2019 ! 1919 ! x 1919 \frac{2019!}{1919!}x^{1919}

Ruilin Wang by Ruilin Wang , 45, USA

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1 solution

For each derivative, the current exponent multiplies the result and the exponent is reduced by 1. Since ∏ p = 2019 − 99 2019 + 0 p = 2019 ! 1919 ! \prod _{p=2019-99}^{2019+0} p=\frac{2019!}{1919!} is true, the result is 2019 ! 1919 ! x 1919 \frac{2019!}{1919!}x^{1919} .

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Nice solution!

Ruilin Wang - 1 year, 10 months ago

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