To find the derivative of the function y = ((3x+1)(cos2x))/e^(2x) using the quotient rule, we need to find the derivatives of the numerator and denominator separately.
First, let's find the derivative of the numerator using the product rule:
dy/dx = (d/dx)[(3x+1)(cos2x)]
= (3)(cos2x) + (3x+1)(d/dx)(cos2x)
Using the chain rule, the derivative of cos2x is -2sin2x:
dy/dx = 3cos2x + (3x+1)(-2sin2x)
Next, let's find the derivative of the denominator:
dy/dx = (d/dx)[e^(2x)]
= e^(2x)(d/dx)(2x)
Using the chain rule, the derivative of 2x is 2:
dy/dx = e^(2x)(2)
Now, we can apply the quotient rule:
dy/dx = (denominator * derivative of numerator - numerator * derivative of denominator) / denominator^2
dy/dx = (e^(2x)(2)[(3cos2x) + (3x+1)(-2sin2x)] - [(3x+1)(cos2x)](2e^(2x))) / (e^(2x))^2
Simplifying the expression:
dy/dx = (2e^(2x)[3cos2x - (3x+1)(2sin2x)] - 2(3x+1)(cos2x)e^(2x)) / e^(4x)
dy/dx = (2e^(2x)(3cos2x - (3x+1)(2sin2x)) - 2(3x+1)(cos2x)e^(2x)) / e^(4x)
Therefore, the first derivative of the given function y = ((3x+1)(cos2x))/e^(2x) is (2e^(2x)(3cos2x - (3x+1)(2sin2x)) - 2(3x+1)(cos2x)e^(2x)) / e^(4x).
Using the quotient rule find the first derivative of the function y=((3x+1)(cos2x))/e^2x
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