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Question

# Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. -If f(x) = sin²(2x), then f’(x) = 2(sin 2x)(cos 2x).

Solution

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$f(x) = sin^2(2x)$

$f'(x) = 2sin(2x)^{2-1} \cdot \dfrac{d}{dx} sin(2x)$

$= 2sin(2x) \cdot cos(2x) \cdot \dfrac{d}{dx}(2x)$

$= 4sin(2x) cos(2x)$

We can see from the chain rule, that the derivative shown for the function is missing an extra factor of 2 from the inner-most funciton (2x)

Also the derivative can be simplified by the sin double angle identity.

$4sin(2x)cos(2x) = 2sin(4x)$

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