Question

A point P is moving along the curve whose equation is
y=((x^3)+17)^(1/2)
When P is at (2,5), y is increasing at the rate of 2units/s. How fast is x changing?

Answer 1

i would simply take the derivative in an implicit manner

\[y=(x^3+17)^{1/2}\]
\[y'=\frac{2x^2x'}{2}(x^3+17)^{-1/2}\]
we know y' so lets solve for x'
\[y'=x^2x'(x^3+17)^{-1/2}\]
\[y'x^{-2}(x^3+17)^{1/2}=x'\]
\[2(2)^{-2}(2^3+17)^{1/2}=x'\]
\[2^{-1}(2^3+17)^{1/2}=x'\]