Find the Derivative - d/dx y=sin(sin(sin(x)))

Problem

d(sin(sin(sin(x))))/d(x)

Solution

1. Identify the outer function as sin(u) where u=sin(sin(x))

2. Apply the chain rule by differentiating the outer sine function to get cos(sin(sin(x))) and multiplying by the derivative of the inner function d(sin(sin(x)))/d(x)

3. Apply the chain rule again to the inner derivative d(sin(sin(x)))/d(x) which results in cos(sin(x)) multiplied by the derivative of its inner function d(sin(x))/d(x)

4. Differentiate the innermost function sin(x) to get cos(x)

5. Combine all the resulting factors from the chain rule applications.

Final Answer

d(sin(sin(sin(x))))/d(x)=cos(sin(sin(x)))⋅cos(sin(x))⋅cos(x)

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