Question

How do you differentiate `e^y=e^(5x)-y^2`?

Answer 1

I found: `(dy)/(dx)=(5e^(5x))/(e^y+2y)`

Explanation:

Here you have an Implicit Differentiation where `y` must be considered as a function of `x` and differentiated as well including the term `(dy)/(dx)`.
We get:
`1*e^y(dy)/(dx)=5e^(5x)-2y(dy)/(dx)`
`e^y(dy)/(dx)+2y(dy)/(dx)=5e^(5x)`
collect: `(dy)/(dx)`
`[e^y+2y]*(dy)/(dx)=5e^(5x)`
and finally:
`(dy)/(dx)=(5e^(5x))/(e^y+2y)`