Question

f(x,y) = (xy^4-y)^5
find partial derivative of x

Answer 1

\[\large f(x,y)=(xy^4-y)^5\]Taking the partial derivative with respect to X gives us,\[\large \frac{\partial f}{\partial x}=5(xy^4-y)^4\frac{\partial}{\partial x}(xy^4-y)\]
Just like in Single Variable Calculus, we have to apply the chain rule! :)

Answer 2

So taking the derivative of the outer function is pretty straightforward, just the power rule.

Now when we take the derivative of this inner part, we treat y as a constant, no product rule inside!

Answer 3

\[\large \frac{\partial f}{\partial x}=5(xy^4-y)^4\cdot (1\cdot y^4-0)\]Understand what we did there? :o

Answer 4

yes thx
so would the partial derivative of y be
5(xy^4-y)^4*(4y^3-1)
?