The correct answer is the following:
cos x times abs. value sin x
quantity divided by sin x
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The derivative of the natural log is 1/x, therefore the derivative is 1/cos(x). However, since the value of cos(x) is submitted within the natural log we must use the chain rule. Then, we multiply 1/cos(x) by the derivative of cos(x). We get the answer: -sin(x)/cos(x) which can be simplified into -tan(x).
There is no real significance to sine plus cosine, now sin2(x) + cos2(x) = 1 for any x, where sin2(x) means to take the sign of the number, then square that value.
f(x) = Cos(x) f'(x) = -Sin(x) Conversely f(x) = Sin(x) f'(x) = Cos(x) NB Note the change of signs.
The derivative of ln x is 1/x The derivative of 2ln x is 2(1/x) = 2/x
If x is a function of time, t, then the second derivative of x, with respect to t, is the acceleration in the x direction.