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What is the derivative of : y = (x^2*ln 4x)/(e^x*cos x)

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lxsptter | Student, Undergraduate | (Level 2) Valedictorian

Posted May 7, 2012 at 5:35 PM via web

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What is the derivative of : y = (x^2*ln 4x)/(e^x*cos x)

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted May 7, 2012 at 5:53 PM (Answer #1)

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The derivative of y = `(x^2*ln 4x)/(e^x*cos x)` has to be determined. This requires the use of the quotient rule, product rule and the chain rule.

y = `(x^2*ln 4x)/(e^x*cos x)`

y' = `((x^2*ln 4x)'(e^x*cos x) - (x^2*ln 4x)(e^x*cos x)')/(e^x*cos x)^2`

=> `((2x*ln 4x + (4*x^2)/(4x))(e^x*cos x) - (x^2*ln 4x)(-e^x*sin x + e^x*cos x))/(e^x*cos x)^2`

=> `((2x*ln 4x + x)(e^x*cos x) - (x^2*ln 4x)(-e^x*sin x + e^x*cos x))/(e^x*cos x)^2`

=> `(2x*ln 4x + x)/(e^x*cos x) - ((x^2*ln 4x)(1 - tan x))/(e^x*cos x)`

The derivative of `(x^2*ln 4x)/(e^x*cos x)` is `(2x*ln 4x + x)/(e^x*cos x) - ((x^2*ln 4x)(1 - tan x))/(e^x*cos x)`

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