I am reviewing some concepts and I do not remember the proper way solve this problem.
I need to differentiate f with respect to x and with respect to y (partial derivatives) for:
f(x,y) = ln(x^5+xy+(y^3)(e^x))
My guess is that f with respect to x is:
(5x^4+y+(y^3)(e^x))((1/ (x^5+xy+(y^3)(e^x)))
But I am pretty sure that my guess is not correct. Can anyone help me with this, please?
you can see the attachment for the answer
the mistake you were making is for the implicit differentiation part
for example IF you have to differentiate f(x) = xy
then you use product rule = derivative of the first times the second + the derivative of the second times the first = 1y + y'x when you differentiate y then you get y'
Second example f(x)= (x^3)(y^3) = 3(x^2)(y^3) + 3(y^2)y'(x^3)
you can review implicit differentiation from youtube
I thought that I did not need to add the y part of it while differentiating for x.
I know that the derivative of ln(x) is 1/x, based on that assumption I solved ln(x^5+xy+(y^3)(e^x)) as 1/(x^5+xy+(y^3)(e^x)).
My problem is that I do not know how to link the derivative of the part inside the brackets with the derivative of ln(x).
Because I am treating y as a constant, I took the derivative of the part inside the brackets and connected it with the derivative of ln(x); however I do not know if I am supposed to do this:
((1/ (x^5+xy+(y^3)(e^x)))(5x^4+y+(y^3)(e^x))
Sorry about the wrong information. I have not learned partial derivatives yet in Calculus. I just looked it up on Google and you are right about treating y as a constant. I thought you were talking about implicit differentiation. I am sorry for the misunderstanding.