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6cos(2x) +sin(x).Using the double angle formula for cosine (or otherwise), cos(2x) = cos(x)cos(x) - sin(x)sin(x) .cos(2x) = cos^2(x) - sin^2(x) .Hence, 6cos(2x) +sin(x) = 6(cos^2(x) - sin...

The first part of the problem is solved by differentiating once and equating this to zero:
y = x^3 - 3x^2 +4 .dy/dx = 3x^2 - 6x .dy/dx = x(3x - 6...

The answer can be found by using the chain rule and simple substitution as well as basic knowledge of differentiation.f’(x)= -6cos(3x)sin(3x)

cos^2(A) + sin^2(A) = 1, sin(A+B) = sinAcosB + cosAsinB, cos(A+B) = cosAcosB - sinAsinB, sin2A = 2sinAcosA (if A = B), cos2A = cos^2(A) - sin^2(A) (if A = B), cos2A = 1 - 2sin^2(A) => sin^2(A) = 1/2(1 ...

This equation can be solved using separation of variables. Firstly we rearrange the equation so that all of the y's are on the left hand side and all of the x's are on the right: 1/y²* dy = 6x ...