In Exercises 53 and 54, find dr/ds.

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Question

In Exercises 53 and 54, find dr/ds.

r cos 2s + sin²s = π

Accepted Answer

Welcome back, everyone. By using implicit differentiation, find D X divided by D Y for the curve given by the equation X cosine of 3 Y plus squared of Y equals 4. 1st of all, let's understand that X is a function of Y, so X equals X of Y. So Y is our independent variable while X is the dependent variable. So, to use the implicit differentiation method, we want to differentiate the left hand side, which is X cosine of 3 Y. Plus squared of y. And we want to set this derivative equal to the derivative of the right hand side, so the derivative of 4. As we can see, first of all, we have to differentiate the product X multiplied by cosine of 3 Y. Then we're going to add the derivative of sin squared of y. And this is going to be equal to the derivative of 4, which is just 0 because we have a constant, right? So let's begin by differentiating X cosine of 3 Y. According to the product rule, we are first of all, taking the derivative of X, multiplying that by the cosine of 3 Y. And then we're adding X multiplied by the derivative of cosine of 3 Y. So here we're just applying the product rule, and then the derivative of sine squared of Y, according to the power rule is going to be 2 of Y. Multiplied by the derivative of the inner function right because we have a composite function. The outer function is the square, and now the inner function is sin of y. So we're basically multiplying by the derivative of sine of y, and we're setting it equal to 0. Now, the derivative of X is going to remain as X because we're looking for the X divided by D Y, which is X. We then have cosine of 3 Y. What is the derivative of cosine of 3Y? Well, we're going to get negative sine of 3Y, and according to the chain rule, we're multiplying that by the derivative of the inner function. So the derivative of 3 Y. We are then adding two sine of Y, which is multiplied by the derivative of sine, and the derivative of sine is cosine of Y. We're setting it to 0 once again because that's what we have on the left hand side. And now we want to evaluate. The remaining derivatives. So, first of all, we have X. Cosim Of 3 Y. Now, we can replace the positive sign with the negative sign, and this gives us negative. Ay. We why The derivative of 3 Y is 3 based on the based on the power rule. And then we are adding to sign of Y. Cosine of y. From here, we have to recall that. 2 sine of Y cosine of Y can be written as sign of toy because we have a double angle formula, right? So we're going to replace it with sign of 2Y. And we can say that X cosine of 3 Y minus 3 X 3 Y. Plus of 2y equals 0. If we want to solve for X, we're going to isolate X cosine of 3y on the left, and we're going to move all of the other terms to the right. So this gives us 3 X of 3Y. Minus 2y on the right hand side. Therefore, X is going to be equal to the right hand side, which is 3 X. Sign of 3 Y. Minus sign of Ty. Divided by the factor adjacent to X, so that's cosine. Of 3 Y. And that would be our final answer. We can also label it as D X divided by the Y because that's what X Reaper sounds. So let's label our final answer and thank you for watching.

In Exercises 53 and 54, find dr/ds.

r cos 2s + sin²s = π

Key Concepts

Implicit Differentiation

Chain Rule

Trigonometric Identities

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