Implicit Differentiation

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Question

Implicit Differentiation

In Exercises 43–50, find by implicit differentiation.

xy + 2x + 3y = 1

Accepted Answer

Welcome back, everyone. By using implicit differentiation, find D Y divided by DX for the curve given by the equation X Y plus 4 x minus 2y equals 5. If we want to use implicit differentiation, we have to take the left hand side of the equation, which is XY plus 4 x minus 2y. We're going to differentiate it, and we're going to set it equal to the derivative of 5. So, basically speaking, we're taking the derivative of each side of the equation. Let's go ahead and evaluate each derivative. As we can see, we need the derivative of Xy followed by the derivative of X and the derivative of Y. This is going to be equal to the derivative of 5, which is 0, right, because we're differentiating a constant. So now let's find the derivative of Xy. Here we have a product, so we have to apply the product rule. The product rule says that we're taking the derivative of X multiplied by Y and adding X multiplied by the derivative of Y. Now X is the independent variable and its derivative is one based on the power rule, so we can simplify the derivative of X Y as Y plus XY derivative. Well done. And now we're substituting this expression into the original implicit differentiation. We're going to get Y plus X Y + 4 because the derivative of X is 1 minus 2y derivative is equal to 0. And now we want to leave all of the terms involving Y on the same side, so we have X Y. -2 Y. Equals we're going to bring all of the terms not having y to the right hand side. So this gives us negative. Why? Plus 4, right, we can essentially say that we're bringing Y + 4 to the right hand side. We just have to add a negative sign. Now let's go ahead and factor out Y on the left, which leaves us with X minus 2 as a factor, and this is equal to negative Y + 4. And finally, if we want to solve for Y, we have to divide the left hand side by the right hand side, specifically the factor on the left hand side. So we're going to take a negative Y + 4. And divided by the factor X minus 2 adjacent toy. This gives us the solution for Y, which is essentially the Y divided by DX written in a full notation in terms of differentials. Well then, we have our final answer. Thank you for watching.

Implicit Differentiation

In Exercises 43–50, find by implicit differentiation.

xy + 2x + 3y = 1

Key Concepts

Implicit Differentiation

Chain Rule

Solving for dy/dx

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