### Chapter Three Test, Question Three

3. Differentiate the function: f (theta) = ln(cos 7(theta))

a) f ' (theta) = -7 tan (7(theta))

b) f ' (theta) = sec (7(theta))

c) f ' (theta) = -7 cot (7(theta))

d) f ' (theta) = 7 sec (7(theta))

So, let's differentiate. Because we have three different functions inside of each other, we must use the Chain Rule twice when taking the derivative. For the sake of brevity, I will represent theta using "t".

Step 1) f (t) = ln(cos 7t)

We know that the derivative of ln(x) is simply (1/x). Also, we know that the derivative of cos(x) is -sin(x). Therefore, by the chain rule, we get ...

Step 2) f ' (t) = (1/cos 7t)(-sin 7t)(7)

Now, simplify. Keep in mind that (sin(x) / cos(x)) = tan(x)

Step 3) f ' (t) = -7((sin(7t)/(cos(7t)))

Step 4) f ' (t) = -7 tan(7t)

Thus, the correct answer is choice "A."

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