13-26 Implicit differentiation Carry out the following steps.

Question

a. Use implicit differentiation to find dy/dx.

x⁴+y⁴ = 2;(1,−1)

Accepted Answer

Welcome back, everyone. Given the equation X cubed plus Y cubed equals 9, use implicit differentiation to find the first derivative. So we're given our equation that says X cubed plus Y cubed equals 9. Now, what we want to do is apply implicit differentiation, and this means that we want to take the derivative of the left hand side, so X cub plus Y cubed. So, we are differentiating that and we're going to set this equal to the derivative of the right hand side. Essentially, we have the same equation, but we are applying the first derivative for each side. Now what we want to do is differentiate X cubed because we can distribute the derivative and it tells us that we are differentiating X cubed, adding the derivative of Y cubed, and this must be equal to the derivative of 9. So the derivative of X cubed based on the power rule is going to be equal to 3 x 2 because we bring down the exponent and the original exponent decreases by 1 unit. Now the derivative of Y cubed, this is where we have to be a bit more careful, so we get the same 3 Y squad, but we also have to multiply it by Y and that's because according to the chain rule, if our Function is a function of x and not an independent variable while this essentially means that we have to apply the chain rule, right? So notice that in the previous application of the power rule we had X, which is an independent variable, but Y is a function of X. It's not an independent variable, so we're multiplying by Y. And now the derivative of nine is basically equal to 0 because it is a constant. So we can rewrite our equation as 3 x2 plus 3 Y 2 Y equals 0. And from here, our goal is to solve for Y. Before we do that, we can factor out the 3, and we're going to get 3 M C X2 + Y 2 Y equals 0. We can divide both sides by 3, which essentially means that x2 + y2 y equals 0. And now if we subtract X2d from both sides, we're going to get Y2 Y equals negative x2. And now our goal is to solve for Y we can divide both sides by Y2, which gives us Y equals x 2 divided by Y2. And this is essentially the same as Dy divided by DX, right, because Y is D Y divided by DX. So we can say that Dy divided by D X is equal to negative X2. Divided by y2 which corresponds to our final answer. Thank you for watching.

13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
x⁴+y⁴ = 2;(1,−1)

Key Concepts

Implicit Differentiation

Chain Rule

Evaluating Derivatives at a Point

Watch next