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How do you find #dy/dx# by implicit differentiation given #y=cos^2y#?

Answer 1

Evelyn Agard

Implicit differentiation cannot be done on a single valued equation. The given equation is only true at #y ~~0.641714#

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Answer 2

Elijah Abram

To find ( \frac{dy}{dx} ) by implicit differentiation for ( y = \cos^2(y) ), follow these steps:

Differentiate both sides of the equation with respect to ( x ).
Use the chain rule and the derivative of ( \cos^2(y) ), which is ( -2\cos(y)\sin(y) ).
Isolate ( \frac{dy}{dx} ) to solve for it.

The derivative ( \frac{dy}{dx} ) can be found as follows:

[ \frac{dy}{dx} = \frac{d}{dx}(\cos^2(y)) ] [ = -2\cos(y)\sin(y) \cdot \frac{dy}{dx} ] [ \frac{dy}{dx} + 2\cos(y)\sin(y) \cdot \frac{dy}{dx} = 0 ] [ \frac{dy}{dx}(1 + 2\cos(y)\sin(y)) = 0 ] [ \frac{dy}{dx} = 0 \ \text{or} \ 1 + 2\cos(y)\sin(y) = 0 ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.