Well.. this is the formula to get the derivative of the modulus -
d|f(x)|/dx = [ |f(x)|/f(x) ] * f'(x)
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It's either 1 or -1 when x is not equal to zero; at that point |x| is non-differentiable.
The derivative of ln x is 1/x The derivative of 2ln x is 2(1/x) = 2/x
The derivative of 3cos(x) is -3sin(x). This can be found using the chain rule, which states that the derivative of a composition of functions is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In this case, the derivative of cos(x) is -sin(x), and when multiplied by the constant 3, we get -3sin(x) as the derivative of 3cos(x).
The derivative of csc(x) is -cot(x)csc(x).
derivative of sec2(x)=2tan(x)sec2(x)
the derivative of 3x is 3 the derivative of x cubed is 3 times x squared