13–40. Evaluate the derivative of the following functions.
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Question

13–40. Evaluate the derivative of the following functions.

f(x) = 1/tan^−1(x²+4)

Accepted Answer

Hello. In this video, we are going to be evaluating the following derivative. So we are told that we are given a function F of X equal to 1 divided by cosine inverse of 3 X minus 2. Now this looks like a difficult derivative to take at the beginning, but let's go ahead and break down the function little by little. The first thing to notice is that we are given the function in the form of a rational function. What that means is that if we want to take the derivative, we are going to have to use the quotient rule. Now, the quotient rule allows us to assign the numerator its own function and assign the denominator as its own function. We will call the numerator F of X and set it equal to 1. And we will call the denominator G of X and set it equal to cosine inverse of 3X minus 2. Now, in order to use the quotient rule, we have to take the derivatives of the independent numerator and denominator. The derivative of the numerator F of X is equal to 1. is going to be 0 because the derivative of a constant value is 0. But what about the derivative of Govexx? Well, let's go ahead and quickly recall what the derivative of cosine inverse is. The derivative with respect to X of cosine inverse of X is equal to -1 divided by the square root of 1 minus X2. What this means is that if we take the derivative of cosine inverse, whatever we have for the variable X is going to be our X square term in the square root. But what's tricky about our numerator is that we have a function for X. What that means is that our X squared in this case is going to be 3X minus 2, but then we will have to multiply this by the derivative of the innermost function by using chain rule. So how that derivative is going to look is G of X is equal to -1 divided by the square root of 1 minus the quantity 3X minus 2 squared, then by the chain rule, we will multiply this by the derivative of 3X minus 2, and that's just going to be 3. So the derivative of the denominator is going to simplify to be G X equal to -3 divided by the square root of 1 minus 3X minus 2 quantity squared. So, we have the original numerator, the derivative of the numerator, the original denominator, and the derivative of the denominator. With these four pieces of information, we can now use the quotient rule to find the overall derivative. So that is going to be F of X equal to the original denominator, which is cosine inverse of 3X minus 2 multiplied by the derivative of the numerator, which is 0. Minus the original numerator, which is 1. Multiplied by the derivative of the denominator, which is negative 3 divided. By the square root of 1 minus the quantity, 3 X minus 2 squared. And this will all be divided. By the original denominator squared, which is cosine inverse of 3X minus 2, this quantity squared. Now there's a bit of simplification that we need to do. First, cosine inverse multiplied by 0 will just zero out, and -1 multiplied by the negative derivative of the denominator will just make it positive. So currently, we can simplify the derivative to be positive, 3 divided by the square root of 1 minus 3X minus 2 squad, all divided by cosine inverse. Of 3 X minus 2, this whole quantity squared. Next, what we need to do is we need to simplify the complex fraction. Now, a trick here is that we can just go ahead and combine both of the denominators together, and that will simplify the overall derivative to be 3 divided by the square root. Of 1 minus 3X minus 2 quantity squared. Multiplied by cosine inverse of 3 X minus 2, this whole inverse squared. There is no further way to simplify the derivative, so this is going to be the solution to the overall problem. And with that being said, the final solution to this problem is going to be C. So I hope this video helps you in understanding how to approach this type of derivative, and I will go ahead and see you all in the next video.

13–40. Evaluate the derivative of the following functions.
f(x) = 1/tan^−1(x²+4)

Key Concepts

Derivative

Chain Rule

Inverse Trigonometric Functions

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