The curve C has y = (sin x)/ (x), where x>0. Prove that the x-coordinate of any stationary point of C satisfies the equation: x=tan x

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We have y = (sin x)/ x

=> y = (x^-1) * (sin x)

y' = x^-1 * cos x + sin x * (-1)*x^-2

=> y' = (cos x)/x - (sin x)/x^2

The required value of dy/dx = (cos x)/x - (sin x)/x^2

At a stationary point we have...

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We have y = (sin x)/ x

=> y = (x^-1) * (sin x)

y' = x^-1 * cos x + sin x * (-1)*x^-2

=> y' = (cos x)/x - (sin x)/x^2

The required value of dy/dx = (cos x)/x - (sin x)/x^2

At a stationary point we have dy/dx = (cos x)/x - (sin x)/x^2 = 0

(cos x)/x - (sin x)/x^2 = 0

=> x*(cos x) = sin x

=> x = (sin x)/(cos x)

=> x = tan x.

It is thereby proved that x = tan x at the stationary points of the curve y = (sin x)/x

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