Find the derivative of the following functions.
y = In(e...

Question

Find the derivative of the following functions.

y = In(e^x + e^-x)

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 Y equals the natural log of 2 X plus 2 of negative X. Awesome. So it appears for this particular problem we're ultimately trying to figure out what the derivative of this function of Y is. So now that we know that we're trying to find the derivative of this function of Y, our first step is that we need to recall the following differential rule. We need to recall that the natural log prime of X is equal to 1 divided by X. And we are going to need to apply this rule. We need to apply this basic derivative rule, along with the chain rule. So we also need to recall and use the chain rule. Since because of the fact that this function of Y is a composite function. Awesome. So now our second step that we need to take is we need to apply the chain rule. So as we should recall, the chain rule states that Y, where Y denotes the first derivative of this function of Y, because this prime symbol, which looks like an apostrophe in the power, denotes that this prime symbol once again is the first derivative, and since it's Y, the first derivative of the function of Y. So we could say for the chain rule, Y is equal to Dy by DU multiplied by DU by DX. Fantastic. So now, Using the chain rule, we can go ahead and write that Y is equal to 1 divided by 2 or 2 of X, I should say, so 1 divided by 2 X plus 2 of negative X multiplied by D by DX because we're taking the derivative with respect to X. Of 2 to the power of X plus 2 to the power of negative X. Fantastic. So now our third step is we now need to figure out our two derivatives, so we need to solve for our two derivatives. And as we should recall a note from the table of basic derivatives, A to the power of X is equal to a power of X. So, we also need to note that the natural log of A A has to be greater than 0, and we also need to note that A cannot equal 1. Awesome. So A cannot equal one. So with that in mind, We can write that D by DX of 2 to the power of X is equal to 2 X multiplied by the natural log of 2, fantastic. So once again, D by DX, so taking the derivative with respect to X of 2 power of X, we'll get 2 power of X multiplied by the natural log of 2. And we also need to note that for 2 to the power of negative X, since this is a composite function, we need to recall and apply the chain rule. So for D by DX of 2 to the power of negative X, we need to take this and we need to say that it's equal to 2 power of negative X, multiplied by the natural log of 2, multiplied by D by DX. Of negative X, which is equal to -2 to the power of negative X multiplied by the natural log of 2. Fantastic. So now our 4th step is we need to substitute in our results from step 3 into our expression that we found in step 2, and when we do that, we will find that Y is equal to 1 divided by 2 X plus 2 of negative X multiplied by 2. X multiplied by the natural log of 2 minus 2 to the power of negative x multiplied by the natural log of 2. And when we do some simplifying, we will find that Y is equal to the natural log of 2 multiplied by 2 X minus 2 of negative X, all divided by 2 X plus 2 of negative X. And that is our final answer. Hooray, we did it. So therefore, in conclusion, the derivative of the function of Y is Y is equal to the natural log of 2 multiplied by 2 X minus 2 of negative X, all divided by 2 X plus 2 power of negative X. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Find the derivative of the following functions.
y = In(e^x + e^-x)

Key Concepts

Derivative

Natural Logarithm

Exponential Functions

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