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(3x + 1)⁴

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. Calculate the derivative of the function F of X is equal to the natural log of 2X minus 3 all to the power of 5. Specifically, So it's saying the natural log of 2 X minus 3, where this expression 2X minus 3 is to the power of 5. Awesome. So with that said, what we're ultimately trying to solve for for this particular problem is we're trying to figure out what the derivative of this function F of X is. So, with that said, our first step to help us solve for this particular problem is we need to note that before we do any sort of differentiation or anything prior to any of this, we need to apply the power rule of logarithms. So that's our first step in order to help us even begin the differentiation process before we can even differentiate or find the derivative, we have to recall and apply the power rule of logarithms. So we need to take FX, so F of X is equal to 5 multiplied by the natural log of 2 X minus 3. So, our second step now is we need to note that we've now obtained our composite function, and our so since we've obtained our composite function, as we should recall, we can now state That you is going to be equal to 2 X minus 3. And we also need to note that we can go ahead and write that F of U of X is going to be equal to 5 multiplied by the natural log of u of X. Fantastic. So now our next step is we need to recall and apply the chain rule in order to help us differentiate. So we can go ahead and write that F of X where this so basically this apostrophe that's in or this apostrophe that's in the power, that means it's the first derivative. So, so F, that's what the prime symbol is. And so this means that we're taking the first derivative of F of X. So F of X. is equal to 5 multiplied by the natural log prime of you, multiplied by U. Of X, so, so you subscript X, so it's you subscript X. is equal to 5 divided by U multiplied by U subscript X. So UX subscript X. Awesome. So then we can go ahead and write that F of X is equal to 5 divided by 2 X minus 3 multiplied by D by DX of 2X minus 3. And our third step now is we can now evaluate the final derivative that we found in step 2, so we can write that D by DX of 2 X minus 3 is equal to 2 multiplied by D by DX of X minus D by DX of 3, which will be all equal to 2. And our 4th and final step is we need to substitute the result we got from step 3, and we need to plug it into the result that we found from step 2. And when we do that, we will find that F of X is equal to 5 divided by 2 X minus 3, multiplied by 2. And when we simplify, we will find that F of X is going to be equal to 10 divided by 2 X minus 3, and that is gonna be our final answer. Hooray, we did it. We found our final answer for the derivative. So without a doubt, We can say that our derivative for the function of F of X is going to be F of X is equal to 10 divided by 2 X minus 3. 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(3x + 1)⁴

Key Concepts

Derivative

Logarithmic Properties

Chain Rule

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