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How do you differentiate #f(x)=(xsin2x)/(x^2cos^2x-tanx)# using the quotient rule?

Answer 1

Cameron Alexander

See the answer below:

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Answer 2

Ezekiel Alexander

To differentiate f(x) = (xsin(2x)) / (x^2cos^2x - tan(x)), you can use the quotient rule. The quotient rule states that if you have two functions, u(x) and v(x), then the derivative of their quotient is given by the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Applying the quotient rule to f(x), we get:

f'(x) = ((x^2cos^2(x) - tan(x)) * (2xcos(2x) + sin(2x)) - (xsin(2x)) * (2xcos^2(x) + 2x^2cos(x)sin(x) - sec^2(x))) / (x^2cos^2(x) - tan(x))^2

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.