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).

y = 4 log₃(x²−1)

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Calculate the derivative of the function Y equals 5 multiplied by logba 2 of 3 X2 + 2. Awesome. So ultimately what we're trying to solve for in this particular problem is we're trying to figure out what the derivative of this function of Y is. So now that we're ultimately knowing that now ultimately that we know that we're trying to figure out what the derivative of this function of Y is, our first step in order to solve this problem is we need to recall and use the following equation, which states that D by DX. Of log base A of X is equal to 1 divided by X multiplied by the natural log of A. Awesome. So now we need to apply this basic derivative along with the train rule, since Y is a composite function. So as we should recall a note, when we look at our function of Y, we should be able to recall and recognize the fact that it is a composite function. And because it's a composite function, we're going to need to use the chain rule, and since we're dealing with logs, we need to recall and use our basic derivative equation to help us take the derivative of the log of this particular log expression. So with that said, we are going to we're gonna, that's our first step. So our first step is we need to recall. How to take the derivative of a log, and now our second step that we need to do is we need to recall and apply the chain rule, and we need to factor out the constant prior to our differentiation. So as we should recall and note that when we have a constant in any of our Or any of our functions or equations, we're going to need to factor that out prior to us. So before we even start differentiating, we need to pull out this constant value. So with that said, we need to recall and use the the equation for, or rather the formula for the chain rule, and the chain rule states that Y is equal to Dy by DU multiplied by DU by DX. So when we apply the chain rule, we will find that Y is equal to 5 multiplied by 1 divided by 3 X2 + 2 multiplied by the natural log of 2 multiplied by D by DX of 3 X2 + 2. And now our third step that we need to take is we need to recall that we need to evaluate the final derivative using the power rule, which let us recall that the power rule states that X to the power of N is equal to N multiplied by X of N minus 1. And let us also recall a note that CX is equal to 0, where C denotes a constant. So with that said, we could go ahead and write that D by DX of 3 X2 + 2 is equal to 3 multiplied by D by DX of X2. Plus, so we need to take, once again, we're taking 3 multiplied by D by DX of X squad. And then we're gonna add, so we're gonna add 2 prime, which will be equal to 3 multiplied by 2 X plus 0, which will be all equal to 6 X. Fantastic. So now our 4th and final step is we need to substitute the result from step 3 into our expression, so we need to plug in our result from step 3 into the expression in step 2. So when we do that, we will find that Y is equal to 5 divided by the natural log of 2 multiplied by. 3 X2 + 2 all multiplied by 6 X. and when we simplify this expression, we will find that Y is equal to 30 X divided by the natural log of 2 multiplied by 3 X2 + 2. And this will be our final answer for our derivative. Hooray, we did it. So once again, in conclusion, Our derivative for the function of Y will be Y, which the prime symbol indicates that that's the derivative. So the derivative of the function of Y Y is equal to 30 X, all divided by the natural log of 2 multiplied by 3 X2 + 2. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

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).

y = 4 log₃(x²−1)

Key Concepts

Derivative

Logarithmic Functions

Change of Base Formula

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