Evaluate the derivative of the following functions.
f(x)...

Question

Evaluate the derivative of the following functions.

f(x) = sec^-1(ln x)

Accepted Answer

Welcome back, everyone. Find the derivative of G of X equals in vertical sequence of 6 LN X. We're given 4 answer choices A -1 divided by X, square root of 6 LNX2 minus 1. B says 6 divided by X square root of 1 minus 6 LNX squared. C says 6 divided by X. The absolute value of LNX multiplied by square root of 6 LNX quad minus 1. D says 11 divided by X. The absolute value of LN of X multiplied by square root of 6 LNX square minus 1. So we want to identify the derivative of the function G of X. This means that we want to differentiate. In vertical sequence of 6 m of X. And now what we're going to do is simply show that we are evaluating the derivative. Let's write it more neatly to make sure that we avoid multiple sets of rencies, so that's inverse cosequence of 6L and of X. Now let's recall. The derivative of the inverse cosequent, if we want to evaluate the derivative of inverse cosequent. Based on the table of basic derivatives, this is -1 divided by the absolute value of x multiplied by square root of X2 minus 1. The main difference in this case is that our angle is 6 LN of X and not X, which essentially means that we want to apply the chain rule. So let's go ahead and apply it. First of all, we're going to keep our angle 6LN of X using the basic derivative, right? So we get -1 divided by the absolute value of the angle. So that would be the absolute value of 6 LN of X. Multiplied by square root of the angle squared, so that'd be 6 L and X squad minus 1, and based on the chain rule now we need to multiply that by the derivative of the inner function which is 6 LN of X, so we're multiplying by 6 LN derivative of X. And now let's recall that the derivative of LNX is 1 divided by X. So we're going to use that and overall we're going to get -1 in the numerator. -1 multiplied by 6 gives us 6, so we can actually write 6 in the numerator. Then we're going to get X in the denominator which comes from the derivative of L and of X, right? It is 1 divided by X, so we get that X, and now we can continue with the remaining terms. Also, let's understand that the absolute value. Of 6 is equal to 6, so we can also write 6 in front of X, which leaves us with the absolute value of LN of X, and then under the radical we can write 6 ln of X squad minus 1. Notice that now we can cancel out those sixes which will give us our simplified answer and that's -1 divided by the absolute value. Of L and of X. We also have X in front. And then we can finish with the radical term. So we're going to write. Square root Of 6 LNX squared minus 1. Of course we can square it and say that this is 36 ln squared X, but this is not necessary. Now looking at the answer choices, we can conclude that the correct answer to this problem corresponds to the answer choice and D. Thank you for watching.

Evaluate the derivative of the following functions.
f(x) = sec^-1(ln x)

Key Concepts

Derivative

Inverse Functions

Chain Rule

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