Find the slope m of the curve $x + y^3(x)= 6$ at the point $(1, \sqrt[3]5)$

Question

When asking if y(x) is equal to the y value I'm told that " y(x) is the y value, but viewed as a function of x in this context, not as an independent variable."

What exactly does this mean? Should I replace with y(x) with 6−x or should I just differentiate the original function normally using implicit differentitation?

Answer 1

Your function is $y=\sqrt[3]{6-x}$ You can differentiate that or you can implicitly differentiate the original equation getting $1+3y^2y'=0$ Since you know $y$ you can compute $y'$.

Answer 2

Implicitly differentiate your function, then insert the point coordinates $(x,y)=(1,\root3\of{5})$ into it.

Alternatively, you can just isolate $y$ and differentiate as usual.