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Calculus/Derivative and evaluate

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Expert: - 6/3/2009


Question
find the derivative and evaluate f(x)= (1+sin(x))/(tan(x)) at X=pi/6  

Answer
Before taking the derivative, it is best to simplify the expression first and therby drop any parenthisis.

Since we have f(x) = (1+sin(x))/(tan(x)),
remember that tan(x) = sin(x)/cos(x).

Since the we are dividing by tan(x), we can multiply by the inverse.

We now have f(x) = (1+sin(x))(cos(x)/sin(x)).

Multiplying this out gives us cos(x)/sin(x) - cos(x).

We next remember that cos(x)/sin(x) = ctn(x), so now
f(x) = ctn(x) - cos(x).

From here, f'(x) can be easily found.
It is just cscē(x) + sin(x).

To evaluate it at π/6 radians, not that π/6 radians is 30°.
That is a common triangle with the near side is √3, the far side is 1, and the hypoteneuse is 2.

We know that the sin(x) = 1/2.
We know that csc(x) = 1/sin(x) = 1/(1/2) = 2, so cscē(x) = 4.
Thus the answer if f'(π/6) = 1/2 + 4 = 4.5, right?

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