How do you use implicit differentiation to find dy/dx given #xe^y-y=5#?
1 Answer
# dy/dx = (e^y)/(1-xe^y)#
Explanation:
When we differentiate
However, we cannot differentiate a non implicit function of
When this is done in situ it is known as implicit differentiation.
We have:
# xe^y-y=5 #
Differentiate wrt
# (x)(e^ydy/dx)+(1)(e^y) - dy/dx=0#
# :. xe^ydy/dx + e^y - dy/dx=0#
# :. e^y = dy/dx-xe^ydy/dx#
# :. (1-xe^y)dy/dx = e^y#
# :. dy/dx = e^y/(1-xe^y) #
Advanced Calculus
There is another (often faster) approach using partial derivatives. Suppose we cannot find
# (partial F)/(partial x) (1) + (partial F)/(partial y) dy/dx = 0 => dy/dx = −((partial F)/(partial x)) / ((partial F)/(partial y)) #
So Let
#(partial F)/(partial x) = e^y#
#(partial F)/(partial y) = xe^y-1 #
And so:
# dy/dx = -(e^y)/(xe^y-1) = (e^y)/(1-xe^y)# , as before.