9–61. Evaluate and simplify y'.

y = 2x² ...

Question

9–61. Evaluate and simplify y'.

y = 2x² cos^−1 x+ sin^−1 x

Accepted Answer

Hello. In this video, we are going to be determining the derivative of the given function. The problem tells us that the function Y is defined as 3 X 2 multiplied by sine inverse of X, plus cosine inverse of X. So how can we take the derivative of this function? Well, since we have a sum of multiple terms, in order to take the derivative, we will first need to take the derivatives of the individual terms. Let's label the first term as term 1 and the second one as term 2. Let's start by taking the derivative of the first term. Now, the first term is given to us as 3 X squared multiplied by sine inverse of X. Since the first term is in the form of the product of two functions, we will take the derivative by using the product rule. The product rule states the following. If we want to take the derivative of the product of two functions, F and G, then that derivative will be defined as F multiplied by the derivative of G plus G multiplied by the derivative of F. So we will label 3 X squad as our F function, and sine inverse of X as our G function. Let's go ahead and take the derivative of our F function. The derivative of 3 X squad by the power rule is going to be 3 multiplied by 2 X, and this will simplify to give us 6 X. By definition, the derivative. Of sine inverse of X. is given to us as one divided. By positive, square root 1 minus X squared. Since we have the 4 pieces of information we need for the product rule, let's go ahead and take these pieces and take the derivative of the first term. So by the product rule, we will take the first function, 3 x 2d and multiply this by the derivative of the second function, which is 1 divided by the square root of 1 minus X2. Then add the second function, sign inverse of X, and multiply this by the derivative of the first function, which is 6 X. Now, let's go ahead and simplify. We will multiply the first two terms together, and that will leave us with 3 X squared, divided by the square root of 1 minus X2. And sine inverse of X, multiplied by 6 X will simplify to be 6 X sign inverse of X. This is going to be the derivative of the first term. Now what about the derivative of the second term? Well, by definition, the derivative. Of cosine inverse of X is defined as -1 divided by the square root of 1 minus X2. So this is going to be the derivative of the second term. Now that we have the derivative of the 1st and 2nd term, let's go ahead and place them back together. That is going to give us Y is equal to 3 X 2 divided by the square root of 1 minus X2 plus 6 X multiplied by sin inverse of X minus 1 divided by the square root of 1 minus X2. Notice that the first and last term of the derivative have the same denominator. That means that we can go ahead and combine these terms together. That is going to allow us to simplify the overall derivative to be 6 X sin inverse of X. Plus 3 x 2 minus 1 divided by the square root of 1 minus X2, and this is going to be the simplest version of the derivative of the original function. And with that being said, the solution to this problem is going to be a. So I hope this video helps you in understanding how to calculate this type of derivative, and I will go ahead and see you all in the next video.

9–61. Evaluate and simplify y'.y = 2x² cos^−1 x+ sin^−1 x

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9–61. Evaluate and simplify y'.

y = 2x² cos^−1 x+ sin^−1 x