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.

49. 3arctan(x) + arcsin(y) = π/4; P(1, -1)

Accepted Answer

Welcome back everyone. Use implicit differentiation to find the Dy divided by dx at the point p. 0.1/22 given that inverse of tangent of x divided by 2 + 3 inverse of sin of y is equal to pi divided by 2, a square root of 3 divided by 12, b negative square root of 3 divided by 12, c negative square root of 3, and d 0. So for this problem, we want to apply implicit differentiation, which essentially means that we're going to differentiate the left hand side and the right hand side. So let's start with each derivative individually. We're going to differentiate inverse of tangent of x divided by 2 1st. And we want to apply the chain rule because we have a composite function. So first of all, we're differentiating inverse of tangent with respect to x divided by 2. Let's remember that. The derivative of inverse of tangent is 1 divided by 1 plus the square of the inner function, and our inner function is x divided by 2. And then according to the chain rule we're multiplying by the derivative of x divided by 2. Here we can factor out one half and differentiate x. The derivative of x is 1. So we basically get 1/22 right for the derivative multiplied by 1 divided by 1 + x2 divided by 4. And now let's identify the derivative of 3 inverse of sin of a y. We can factor out the constant which is 3, and we can differentiate the inverse of sin of y. Again, we have a composite function because y is a function of x. So we get 3 multiplied by 1 divided by the square root of 1 minus the square of the inner function. So that would be y squared, and according to the chain rule we are multiplying by the derivative of y. We can just label it as d y. Divided by the x, right? Because Y is a function of x. And now on the right hand side we have pi divided by 2. The derivative of a constant is 0. So what we can do is simply take our left hand side, we can basically label it as 1 divided by 2 multiplied by 1 gives us 2 plus 2 multiplied by x 2 divided by 4 gives us x 2 divided by 2. And we're going to add. 3 divided by square root of 1 minus y2 multiplied by Dy divided by dx, this is going to be equal to 0. And then we want to solve for dyy divided by dx so we can subtract 1 divided by 2 plus x 2 divided by 2 from both sides, and then we're going to divide by 3 divided by the square root of 1 minus y2. Let's go ahead and do that. 3 divided by the square root of. 1 minus y2 multiplied by Dy divided by dx is going to be equal to 1 divided by 2 + x2 divided by 2. And now let's multiply both sides by square root of. 1 minus y2 divided by 3, which essentially allows us to isolate the derivative, right. Well done. So now that we have our derivative, we want to evaluate that derivative at X equals 0, Y equals 1/22. Let's go ahead and do that. So we are going to show that Dy divided by dx at. X equals 0. Y equals 1/2 is going to be equal to. And we're going to use the expression, right? 4 d y divided by d x we're going to substitute x equals 0 y equals 1/2, so that'll be negative 1 divided by 2 + 02 divided by 2 multiplied by a square root of 1 minus y2, so that'd be 1/2 square divided by 3. Now, all that we have to do is simplify. So we get -1 divided by 2 + 0, that's just 1/2 multiplied by the square root of. 1 minus 1/4, that's square root of 3/4, or basically square root of 3 divided by 2, right? Square root of 3 divided by 2. And divided by 3. So the square root of 3 is divided by 2 multiplied by 3, or basically 6. Well then, in our multiplying we get negative square root of 3 divided by 12. So now considering the answer choices. We're going to conclude that the correct answer for this problem corresponds to the answer of choice B. Thank you for watching.

For problems 49–52 use implicit differentiation to find dy/dx at the given point P.49. 3arctan(x) + arcsin(y) = π/4; P(1, -1)

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For problems 49–52 use implicit differentiation to find dy/dx at the given point P.
49. 3arctan(x) + arcsin(y) = π/4; P(1, -1)