Differentiating Implicitly

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Question

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x³ + y³ = 18xy

Accepted Answer

Welcome back everyone. Using implicit differentiation, what is the D Y divided by D X of the function below? X cubed plus y cubed equals 15 x y. If we want to use implicit differentiation, this means that we're differentiating the left hand side of the equation and setting it equal to the derivative of the right hand side. So let's perform differentiation. We want to differentiate the X cubed. We want to add the derivative of Y cubed and set it equal to 15 multiplied by the derivative of XY. Now let's identify each of the three derivatives. Starting with X cubed, we can basically apply the power rule, which gives us 3 X2. Now, let's differentiate Y cubed. We have to understand that this is a composite function because Y is a function of X. So first of all, we apply the power rule, which gives us 3 Y squared. And according to the chain rule, we also have to multiply by Y, which is the derivative of Y with respect to X. And finally for the derivative of XY. Let's apply the product rule. We're taking X, multiplying by Y, and adding X multiplied by Y. And we also know that the derivative of X is 1. This is our independent variable. So this simplifies to Y + XY. And now let's substitute all of those into the previous expression, which results in 3 X2 + 3 Y 2 Y. Equals 15. Emre why. Plus xy. We can divide both sides by 3. And get x2 + y2 Y equals. 5 Y plus 5 XY we have also distributed the terms on the right hand side. And now what we want to do is simply solve for Y. So we are going to. Take on the left hand side and factor it out, which gives us Y squared. Now what we notice is that we have on the right, so we're going to move that term to the left and it is going to become negative. So factoring out from 5 X Y gives us 5 Xs with a negative sign on the left. So we got Y C Y2 minus 5 X. And this is going to be equal to 5 Y. And now we're moving X2 to the right, which gives us negative X2. From here we can solve for Y by dividing. The right hand side, 5 Y. Minus X2 by the factor on the left hand side which is Y2 minus 5 X. And we have determined the derivative Dy divided by D X, which is our final answer. Well done. Thank you for watching.

Differentiating Implicitly

Use implicit differentiation to find dy/dx in Exercises 1–14.

x³ + y³ = 18xy

Key Concepts

Implicit Differentiation

Chain Rule

Solving for dy/dx

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