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Chapter 6: Problem 84

If \(y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots \text { to } \infty}}}\) then find \(\frac{d y}{d x}\).

Short Answer

Expert verified
The derivative \( \frac{dy}{dx} \) of the function \( y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\ldots to \infty}}} \) is \( -\frac{\sin x}{2y - 1} \).

Step by step solution

01

Simplify the infinite nested radicals

Since the nesting of radicals is infinite, we can let \( y = \sqrt{\cos x + y} \). This equality is due to the nature of the infinite breeding pattern. Treating \( y \) as a recurrent factor, we simplify this equation.
02

Solve the equation for \( y^2 \)

Solve \( y = \sqrt{\cos x + y} \) for \( y^2 \). This leads us to an equation \( y^2 = \cos x+y \). Rearranging terms, we get \( y^2-y=\cos x \).
03

Differentiate the equation

Now, we differentiate the equation \( y^2-y=\cos x \) with respect to \( x \). Using the Chain Rule for differentiation, we attain \( 2y\frac{dy}{dx} - \frac{dy}{dx} = -\sin x \).
04

Solve for \( \frac{dy}{dx} \)

On rearranging Step 3, we get \( \frac{dy}{dx}(2y - 1) = -\sin x \). Finally, we will solve this equation for \( \frac{dy}{dx} \) which gives \( \frac{dy}{dx} = -\frac{\sin x}{2y - 1} \)

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