What is the implicit derivative of #5=x-1/(xy^2)#?
When finding the implicit derivative of an equation, we are finding the derivative in respect to
This means that when we find the derivative of any variable other than
Knowing this, we will start with implicit differentiation.
Here is what we are given:
We can go ahead and avoid the Quotient Rule by multiply each term by
Here we can use the Product Rule to solve:
Please see below.
I assume that we want
To avoid the quotient rule, I would start by multiplying both sides be
Differentiating with respect to
Now we'll do some algebra.
# = ( 2xy - 5y)/(10x - 2x^2)#
To see that these are equivalent, start with
# = ( 2xy - (5)y)/(2x(5) - 2x^2)#
# = ( 2xy - (x-1/(xy^2))y)/(2x(x-1/(xy^2)) - 2x^2)#
Simplify algebraically to get
# = - (x^2y^3+y)/(2x)#
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