Section 3.10 : Implicit Differentiation
1. For xy3=1 do each of the following.
- Find y′ by solving the equation for y and differentiating directly.
- Find y′ by implicit differentiation.
- Check that the derivatives in (a) and (b) are the same.
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a Find y′ by solving the equation for y and differentiating directly. Show All Steps Hide All StepsStart Solution
First, we just need to solve the equation for y.
y3=x⇒y=x13 Show Step 2Now differentiate with respect to x.
y′=13x−23Start Solution
First, we just need to take the derivative of everything with respect to x and we’ll need to recall that y is really y(x) and so we’ll need to use the Chain Rule when taking the derivative of terms involving y.
Also, prior to taking the derivative a little rewrite might make this a little easier.
xy−3=1Now take the derivative and don’t forget that we actually have a product of functions of x here and so we’ll need to use the Product Rule when differentiating the left side.
y−3−3xy−4y′=0 Show Step 2Finally, all we need to do is solve this for y′.
y′=y−33xy−4=y3xFrom (a) we have a formula for y written explicitly as a function of x so plug that into the derivative we found in (b) and, with a little simplification/work, show that we get the same derivative as we got in (a).
y′=y3x=x133x=13x−23So, we got the same derivative as we should.