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

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Answer 1 · 2015-01-26T09:03:02

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))#