Repeated Derivatives

Level 2

If you take the derivative of $X^{N}$ repeatedly, what is the last non-zero term of the series? Enter your answer evaluated at N=48 and state the answer as $A$ in the equation $Y\times 10^{A}$ .

Note: $Y$ is a not necessarily an integer and $Y\times 10^{A}$ is scientific notation.

The answer is 61.

1 solution

Daniel Xian
Jan 27, 2019

If you take the repeated derivative of $X^{4}$ you will find that the exponent, 4, will decrease by one with every derivative, leaving the final term as $4 \times (4-1) \times (4-2) \times (4-3)$ . This is exactly the definition of the Factorial Function. So we find 48!, which is approximately $1.24 \times 10^{61}$ which means the answer is $\boxed{A=61}$ .

You should specify that Y is not an integer. The way the question looked, I assumed you just wanted the count of zeros at the end of the factorial, which is 10.

Vadim Evstifeev - 2 years, 4 months ago

Thanks! Changed that.

Daniel Xian - 2 years, 4 months ago

Repeated Derivatives

The answer is 61.

1 solution

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