Find dy/dx by implicit differentiation : e^(x/y) = x - y

Question

Find dy/dx by implicit differentiation : e^(x/y) = x - y

Answer 1

Implicit differentiation here asks that we differentiate both sides with respect to x (using differential calculus) and then isolate dy/dx (just using algebra).

 

e^(x/y) = x - y

d/dx [ e^(x/y) ] = d/dx [ x - y ]

 

For d/dx [ x - y ] it is fairly simple: the derivative of x with respect to x is just 1, and the derivative of y with respect to x is dy/dx. So we get:

 

d/dx [ x - y ] = 1 - dy/dx

 

For d/dx [ e^(x/y) ], we have to apply the chain rule. Remember that the chain rule is d/dx g(f(x)) = g'(f(x)) * f'(x). Here the outside function is e raised to a power (whose derivative happens to be the same function), and the inside function is a quotient x/y.

 

d/dx [ e^(x/y) ] = e^(x/y) * d/dx[ x/y ]

 

Remember the quotient rule also, where the derivative of x is just 1 and the derivative of y is dy/dx. We get:

 

d/dx[ x/y ] = ( y * 1 - x * dy/dx ) / y^2

 

Thus our previous equation d/dx [ e^(x/y) ] = d/dx [ x - y ], after differentiation, is:

 

 e^(x/y) * ( y - x * dy/dx ) / y^2 = 1 - dy/dx

 

From here you need to use algebra to isolate dy/dx on one side of the equation. Be aware that you may have both x and y (but not dy/dx) on the other side of the equation, and that is acceptable. Generally you will need to group the terms that have dy/dx on one side of the equation, factor the dy/dx out of those terms, and then divide in order to get dy/dx expressed as a function of x and y.