Calculate the derivative of the following functions. In some c...

Question

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In(sec⁴x tan² x)

Accepted Answer

Welcome back, everyone. Calculate the derivative of the function F of X equals LN of cosequent cubed of X multiplied by cotangent of X. We're given 4 answer choices A says -3, cotangent X minus 2 X. Sent X and B-3 cotent X minus s x tangent X C 3 cotent X minus cos 2 X, and D-3 cotent X plus cosequent X tangent X. So first of all, let's notice that F of X is the natural log function and we're going to use the rules of logarithms. Let's notice that we have a product. So we can rewrite F of X as LN of cosequent cubed of X plus LN of cotangent X. So this is the product rule of logarithms. And now we, we are going to apply the power rule so we can rewrite it as 3 LN of cosequent of X plus LN of cotangent of X, right? So this is the first thing that we can do, and from here we can start differentiating. So let's write down that F of X, the derivative that we're looking for consists of 3 multiplied by the derivative of LN of cosequant of X plus the derivative of Ln. Of cotangent of X. And now we have to recall that the derivative of a natural log is 1 divided by X if we take a of X, right? So in this case, we're going to get 3 multiplied by 1. Divided by cosequant of X, and that's because we have an inner function which is not X as cosequant of X. And now applying the chain rule, we have to multiply by the derivative of cosequant. And we're going to use the same logic for the second function. We're going to take 1 divided by cotangent of X and multiply by the derivative of cotangent of X based on the chain rule. So now we got 3 divided by cosequent of facts. Multiplied by what is the derivative of cosequent well using the table of basic derivatives, that's negative cosequant of X, cotangent of X, and now for the next one, we have to. Take 1 divided by cotangent of X and multiply that by the derivative of cotangent, which is negative cosequent squad of x. So we are nearly done, and now we can begin simplifying. Let's notice that we can cancel out coequan for the first term. For the second term, we have negative cosequence square divided by cotangents. We're going to handle it in a moment. For the first term, we're going to include the negative sign followed by 3, followed by cotangent of X. That's how we get 3 cotangent of X in the answer choices, and then we are subtracting. One Divided by cotangent of X, which is essentially tangent. Multiplied by cosicon squared. Now, what is cosican squared? Well, it's basically one divided by sin squared of X. And this is where we can simplify. So we're going to get -3, cotangent of X. We're going to rewrite tangent as sin of x divided by cosine of X. And this is multiplied by 1 divided by sin squared of X. So here we can cancel out the sign, we can cancel out sine squared, and we're left with 1 divided by cosine, which is sequent X, and 1 divided by sine, which is cosequant of X. So our final answer would be negative. 3 cotangent of X minus 2 X. Go see and a. That's our final answer. Now, let's consider the answer choices. We have 3 cotangent of X minus 2 X cos seek and X, and this corresponds to the answer choice A. Thank you for watching.

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In(sec⁴x tan² x)

Key Concepts

Derivative

Logarithmic Properties

Chain Rule

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