Use logarithmic differentiation to find the derivative of y=(cos2x)^x.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The derivative `dy/dx` has to be determined of `y=(cos 2x)^x`

`y=(cos 2x)^x`

take the natural log of both the sides

`ln y = ln (cos 2x)^x`

=> `ln y = x*ln(cos 2x)`

Differentiate the two sides

`(1/y)*y' = ln(cos 2x) - (2x* sin 2x)/(cos 2x)`

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

The derivative `dy/dx = ln(cos 2x)*(cos 2x)^x - (2x*sin 2x*(cos 2x)^x)/(cos 2x)`

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