Use the power/chain rules:
d/dx (2 cos2 x) = 2 d/dx (cos x)2 = (4 cos x)*d/dx(cos x) = -4 cos x sin x = -2 sin 2x
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d/dx(X^4) = 4X^3 ( first derivative ) d/dx(4X^3) = 12X^2 ( second derivative )
Take the derivative term by term. d/dx(X - cosX) = sin(X) ======
The definition of the derivative, at a point X = x is the limit, as dx tends to 0, of [f(x+dx)-f(x)]/dx. In this case, therefore, it is lim[3*(x+dx)+2 - (3*x+2)]/dx = lim[3x + 3*dx +2 - 3x - 2]/dx = lim[3*dx/dx] = lim[3] = 3.
d/dx ∫ f(x) dx = f(x)
d/dx of lnx is 1/x Therefore the derivative is 1/(1+x)