Question

What is the 20th derivative of x∧-2?

Answer 1

`(d^20y)/(dx^20)x^-2=x^-22xx21!`

Explanation:

When you take the derivative of a negative exponent, the exponent continues to grow:

`(dy)/(dx)x^-2=-2x^-3`

Carrying this out to 20 derivatives...

`(d^20y)/(dx^20)x^-2=(-21xx-20xx-19xx-18xx-17xx-16xx-15xx-14xx-13xx-12xx-11xx-10xx-9xx-8xx-7xx-6xx-5xx-4xx-3xx-2)xxx^-22`

Instead of writing out all of those multiplications, we can express it as kind of a factorial:

`(-21xx-20xx-19xx...-4xx-3xx-2)=(-(21!))/(-1)`

`(cancel(-)(21!))/cancel(-1) = 21! ~=5.1091xx10^19`

now we have the coefficient and we can express the 20th derivative:

`color(green)((d^20y)/(dx^20)x^-2=x^-22xx21!`