Question

What is the derivative of `sqrt(200-x^3)`?

Answer 1

To compute the derivative of the function `f(x)=sqrt(200-x^3)` we can apply the chain rule (in both Lagrange's and Leibniz's notations):

`[h(k(x))]'=h'(k(x)) * k'(x)`

`d/dx h(k(x))=d/dy h(y) |_{y=k(x)} * d/dx k(x)`

I'll adopt Lagrange's notation. In our case `h(y)=sqrt(y)` and `k(x)=200-x^3`, so

`h'(y)=1/(2sqrt{y})`
`k'(x)=-3x^2`

By chain rule:

`f'(x)=[h(k(x))]'=h'(k(x)) * k'(x)=1/(2sqrt{200-x^3})*(-3x^2)=-3/2 x^2/(sqrt(200-x^3))`