What is derivative of y = ln(x^2-sin(e^(2x)))?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The function provided by the problem is a composed function, hence, you need to use chain rule to evaluate the derivative of the given function.

Hence, differentiating with respect to x yields:

`y' = (ln(x^2 - sin(e^(2x))))'`

`y' = 1/(x^2 - sin(e^(2x)))*(x^2 - sin(e^(2x)))'`

`y' = ((x^2)' - (sin(e^(2x)))')/(x^2 - sin(e^(2x)))`

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

Factoring out 2 yields:

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

Hence, evaluating the derivative of the given function, using the chain rule, yields `y' = 2(x - e^(2x)*cos(e^(2x)))/(x^2 - sin(e^(2x))).`

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