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 2x/(x² + 1)³

Accepted Answer

Below there, today we're going to 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. Determine the derivative of the function F of X is equal to the natural log of 5 x divided by 3 x 2 + 4 of 5. Awesome, so it appears for this particular problem, we're asked to solve for the derivative of this function of F of X. So now that we know that we're ultimately trying to figure out what the derivative of F of X is, Our first step that we need to take in order to solve this problem and prior to performing any differentiation processes, we need to first recall and apply the rules of logarithms for the quotient, product, and power, meaning we need to state that. F of X is equal to the natural log of 5 X minus the natural log of. 3 X2 + 4 to the power of 5. And we can also go ahead and write that F of X is equal to the natural log of 5, plus the natural log of X minus 5 multiplied by the natural log of. 3 X2 + 4. And this is what we can write because we can simplify our above function to simplify it down to F of X is equal to this expression, which is the natural log of 5 plus the natural log of X minus 5 multiplied by the natural log of 3 X to the power of 2 + 4. Fantastic. So, now, our second step. That we need to take in order to solve this problem is we need to write down the derivative applying the differentiation to each resultant function. So we need to state that F of X, where this prime, which is like the apostrophe in the power, this indicates the first derivative. So F of X is the first derivative of F of X. And F of X is equal to the natural log of 5 plus the natural log of X minus 5 multiplied by the natural log of 3 X2 + 4. Fantastic. So, our next step now. Which for step 3, is we need to recall and use the table of basic derivatives, and we should be able to recall and note that C subscript X, so CX C subscript X. is equal to 0, and we should also recall that the natural log prime. of X is equal to 1 divided by X. So, due to that, we should be able to evaluate the first two derivatives as follows. We're calling these two. Basics from our derivatives are basically these two rules, to be specific. So applying these two rules, we can go ahead and write that the natural log of 5 is equal to 05 is some constant value. That's what the C denotes as some constant variable, some constant value. So LN 5 is equal to 0. And we also need to note that the natural log prime of X is equal to 1 divided by X. Fantastic. So now, our next step. For step 4, we need to now at this point need to recall a note that for the final derivative we need to recall and apply the chain rule since we are dealing with a composite function, so we need to recall a note that when we look at our function initially, we need to recognize the fact that it's a composite function. And once we've identified it as a composite function, we should therefore note since we're dealing with the composite function, we're going to have to recall and apply the chain rule as a result. So, we also, for this particular problem, we also need to recall and apply the power rule, which, as we should recall, the power rule states that X to the power of N. is equal to N multiplied by X to the power of N minus 1. Fantastic. So with that said, We need to go ahead and write that the natural log of 3x2 + 4 is equal to D by D3X2. Plus 4. Multiplied by the natural log of 3 X2 + 4. Multiplied by D by D X. Of 3 X2 + 4, fantastic. And we go ahead and write that the natural log prime of 3 X2 + 4 is equal to 1 divided by 3 X2 + 4 multiplied by 6 X, and we go ahead and write that the natural log prime. is equal to 6 X divided by 3 X 2 + 4. And now for our fifth and final step, We need to combine all the results from steps 3 and 4 and plug them into our expression given in step 2, meaning we need to go ahead and write that F of X is equal to 0 + 1 divided by X minus 5 multiplied by 6 X divided by 3 X 2 + 4. And we can begin to simplify, and we could write that F of X is equal to 1 divided by X minus 30 X divided by 3 x 2 + 4. We could keep simplifying, we could write that F of X is equal to 3 X2 + 4 minus 30 x 2, all divided by X multiplied by 3 x 2 + 4, fantastic. And we can simplify one last time. And when we do, we will find that F of X, our first derivative of F of X is equal to 4 minus 27 X to the power of 2, all divided by X multiplied by 3 x 2 + 4, and that is our final answer. Hooray, we did it. We found our final derivative. So without a doubt, we can go ahead and state that our final answer for the first derivative of the function of F of X is 4 minus 27 x 2 all divided by X multiplied by 3 x 2 + 4. 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).
f(x) = In 2x/(x² + 1)³

Key Concepts

Derivative

Logarithmic Properties

Quotient Rule

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