Find d/dx(ln√x²+1).

Question

Find d/dx(ln√x²+1).

Accepted Answer

Hello 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. Find the derivative of the function F of X is equal to the natural log of the cube root of X to the power of 3 plus 3X. Fantastic. So it appears for this particular problem, we're ultimately trying to figure out the derivative of this function of F of X. So in order to do that, our first step that we need to take is we need to simplify the given function prior to any differentiation that we're about to do using the power rule of logarithms. So once again, we're trying to recall and use the power rule of logarithms to help us simplify our given function in order to help us make the differentiation a lot easier. So we need to write that the natural log of X to the power of A is equal to a multiplied by the natural log of X. So therefore, as a result, we can go ahead and write that the natural log of the cube root of X to the power of 3 plus 3 X. will be all equal to the natural log of X to the power of 3 + 3 X to the power of 1/3. Which will be all equal to 1/3 multiplied by the natural log of X to the power of 3 + 3 X. Fantastic. And our second step is we need to recall a note and also recognize that the given function is a logamithic function that consists of an outer function which will define as Y equals. One third multiplied by the natural log of you, which you could put you in parentheses or you can write it without, it's whatever you prefer. So we can say that the outer function once again is Y is equal to 1/3 multiplied by the natural log of U. And the inner function, which we'll define as U is equal to X to the power of 3 plus 3X, and we could write this. Because, as a result, that therefore, we will be required to use the chain rule to help us differentiate this particular prompt. So we're able to have our inner and outer functions because we're using the chain rule to help us differentiate this function of FF X. So now, our third step that we're trying to do is we're trying to recall and apply the chain rule to differentiate F of X. So let us recall that F of X, where this apostrophe symbol in the power means prime, and it's F of X means that we're taking the first derivative of F of X, that's what that prime symbol means. So F of X is equal to Dy by DU multiplied by DU by DX. And this will be equal to D by DU of 1/3 multiplied by the natural log of U. fantastic. And we need to multiply this by D by DX of X to the power of 3 plus 3 X, noting that this D by DX means we're differentiating with respect to X, and D by DU means we're differentiating with respect to you. Fantastic. And this will be all equal to 1/3 multiplied by D by DU. Of the natural log of you multiplied by D by DX of X to the power of 3 plus 3 X. Fantastic. Our fourth step is we need to recall and use our table of basic derivatives and the power rule to perform this differentiation, and we need to recall a note that we know that D by DX of the natural log of X. Is equal to 1 divided by X. And we also know that we should note and recall that D by DX of X to the power of N is equal to N multiplied by X to the power of N minus 1. Fantastic. So now, our 5th step that we need to take is we need to combine the rules from step 4 in order to evaluate each resultant derivative. So we can go ahead and write that F of X is equal to 1/3 multiplied by 1 divided by U, multiplied by 3 X to the power of 2 + 3, fantastic because the derivative of 3X will be simply 3. Fantastic. So now our 6th and final steps, so this is our final step. Step 6 is our final step. We need to perform the substitution and note that our substitution is for you. For X 3 + 3 X as U approaches X 3 + 3 X in order to obtain the result in terms of X. So when we do that, we need to take F of X is equal to 1/3 multiplied by 1 divided by X to the power of 3. Plus 3 X multiplied by 3 X to the power 2 + 3. And when we simplify, we will get that F of X, and once again, this is in its most simple form, where we can't simplify any further. So F of X. Will be equal to, once again, if we skip a few steps and rewrite in its most simplified form, it'll be equal to X2 + 1. Divided by X to the power of 3 plus 3 X because when you pull out the 3 to the front of our expression and you are you are distributing the 3 out, the 3s will cancel on the numerator denominator, leaving us with X2 + 1 divided by X to the 3 + 3 X. And that is our final answer. Hooray, we did it. This is our final answer for the derivative of our function of F of X. So once again, in conclusion, the derivative of F of X is F of X is equal to x 2 + 1, all divided by X 3 + 3 X. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find d/dx(ln√x²+1).

Key Concepts

Derivative

Natural Logarithm

Chain Rule

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