Find the derivative of `x^(-3//2)` from the first principle .

Explanation

Let `y = x^(-3//2)`.
Let `deltay` be an increment in y, corresponding to an increment `deltax` in x.
Then, `y +deltay = (x+deltax)^(-3//2)`.
`rArr deltay = (x+deltax)^(-3//2) - x^(-3//2)`.
` rArr (deltay)/(deltax) = ({(x+deltax)^(-3//2) - x^(-3//2)})/(deltax)`
`rArr (dy)/(dx) = underset(deltax rarr 0)("lim") (deltay)/(deltax)`
`rArr (dy)/(dx) = underset(deltax rarr0)("lim") (deltay)/(deltax)`
`= underset(deltax rarr 0) ("lim")({(x+deltax)^(-3//2) - x^(-3//2)})/(deltax) = underset(deltax rarr 0)("lim") ({(x+deltax)^(-3//2)-x^(-3//2)})/({(x+deltax) - x})`
`= underset(urarrx)("lim") ((u^(-3//2) - x^(-3//2)))/((u-x))`, where `u = (x+deltax)`
`= - 3/2x^((-3/2-1))" "[ :' underset(urarra)("lim")(u^(n)-a^(n))/(u-a) = na^(n-1)]`
`= - 3/2x^(-5//2)`.
Hence, `d/(dx) (x^(-3//2)) = - 3/2x^(-5//2)`.