Question

How do you differentiate `e^(x / y) = 8x - y`?

Answer 1

Make use of the method of implicit differentiation to obtain
`dy/dx = (8+ye^(x/y))/(1-xe^(x/y))`

Explanation:

Since we cannot obtain y as an explicit function of x, we will use the method of implicit differentiation to find the derivative of y with respect to x.
In so doing, we must bear in mind that y is a function of x.

We rearrange the expression and then take the derivative on both sides using normal rules of differentiation to obtain

`y=8x-e^(x/y)`

`therefore dy/dx = 8 - e^(x/y)*(x*dy/dx-y*1)`

Note we used the quotient rule in the last term since y is a function of x so x/y represents a quotient of 2 functions in x.

Rearranging we get

`dy/dx = (8+ye^(x/y))/(1-xe^(x/y))`