For problems 49–52 use implicit differentiation to find dy/dx ...

Question

For problems 49–52 use implicit differentiation to find dy/dx at the given point P.

51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)

Accepted Answer

Welcome back everyone. Use implicit differentiation to find the dyy divided by d x at the point P 0.0. Given that x + 1 multiplied by inverse of cosine of y equals pi divided by 2, a pi divided by 2, b pi divided by 4, C pi, and D 0. So let's go ahead and use implicit differentiation. What we're going to do is differentiate both sides of our equation. Notice that on the right hand side we have a constant, so the derivative of a constant is 0, and we simply want to focus on the derivative of the left hand side. And here we have a product. We have x + 1 multiplied by inverse of cosine of y, right? So we're going to apply the product rule. According to the product rule, we take the derivative of x + 1. Multiply by inverse of cosine of y and add x + 1 multiplied by the derivative of inverse of cosine of y. Let's find each derivative. The derivative of x + 1 is 1 + 0. That's basically 1. So for the first term we get inverse of cosine of y plus. X + 1 Multiplied by inverse of cosine of y derivative. So this is a composite function because y is a function of x. Let's remember that the derivative of inverse of cosine is negative 1 divided by the square root of 1. Minus the inner function squared, so that would be y 2, and according to the chain rule we're going to multiply by the derivative of the inner function that would be the derivative of y which we will label as the y divided by d x. And we know that this is going to be equal to 0 because on the right hand side we're differentiating pi divided by 2 which gives us the derivative of a constant that is 0. Now, what we want to do is simply evaluate the derivative DY divided by dx. We can use the point x equals 0. Y equals 0, and simply substitute this point into the equation. So what are we going to get? That be inverse of cosine of y, which is inverse of cosine of 0, plus 0 + 1 multiplied by -1 divided by square root of. 1 minus y2, that'll be 02. Multiplied by Dy divided by dx, and this is equal to 0. So now let's evaluate inverse of cosine of 0. This is going to be pi divided by 2 + 0 + 1 is 1, and we're multiplying by -1 divided by 1, right? So essentially we're going to subtract. 1 multiplied by Dy divided by dx. So pi divided by 2 minus Dy divided by dx is equal to 0, and what we can do is simply add the derivative to both hand sides to show that dyy divided by dx. is equal to pi divided by 2, which corresponds to the answer of choice a. Thank you for watching.

For problems 49–52 use implicit differentiation to find dy/dx at the given point P.51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)

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For problems 49–52 use implicit differentiation to find dy/dx at the given point P.
51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)