Differentiating Implicitly

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Question

Differentiating Implicitly

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

x²y + xy² = 6

Accepted Answer

Welcome back everyone. Using implicit differentiation, determined the correct expression for D Y divided by D X. X 2 Y plus Xy cubed is equal to 10. If we want to use implicit differentiation, we essentially have to differentiate the left hand side, X2 Y plus XY cubed. And set the derivative equal to. The derivative of the right hand side, which is the derivative of 10. So now we're distributing the derivative, we're going to get the derivative of X2 Y. Plus, the derivative of X Y Q is equal to the derivative of 10, which is 0, because the derivative of a constant is 0. Now we have a product between X and Y, so this means we need to apply the product rule. For the first product. Let's identify the derivative of X 2 Y. According to the product rule we're taking the derivative of X2, multiplying by Y. And adding x squared, which is multiplied by the derivative of y. The derivative of X where there's 2 X, so we get 2 X Y. Plus x2y. Well done, and now let's differentiate X Y Q. According to the product rule. Let's take the derivative of X. Multiplied by cubed. And add X, which is multiplied by Y cubed prime. Evaluating each derivative, we know that X is 1, so we get Y cubed. Plus X multiplied by. Now Y cubed is a composite function, so we need to apply the power rule which gives us 3 Y2d. And then according to the chain rule, we are also multiplying by Y. Now we're going to substitute all of those back into the original differentiated form. Which gives us 2 X Y plus X squad. Plus why cubed. Plus 3. X Y 2 Y is equal to 0. Now we're going to factor out Y, which gives us Y in C X2. Plus V X Y 2. Equals And now we're going to move the other terms to the right hand side, so 2 XY becomes negative 2XY on the right. And Y cubed becomes negative Y cubed. On the right We can also factor out the negative sign. So that we have a sum within. And now solving for y we simply need to divide the right hand side. Negative. 2 XY plus Y cubed. By the factor on the left hand side adjacent to white prime. And that's X squared. Plus 3 XY squared. And that would be our finally answers this problem we have identified. D Y divided by d X, which is the same as white prime. Thank you for watching.

Differentiating Implicitly

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

x²y + xy² = 6

Key Concepts

Implicit Differentiation

Product Rule

Chain Rule

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