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 - 1)(x + 2)³ / (1 - 4x)²

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 H X equals the natural log of X2 minus 1 multiplied by 4 minus x to the power of 5, all divided by X + 3 to the power of 4. Awesome. So it appears for this particular problem we're asked to solve for the derivative of this function of H of X. So now that we know that we're ultimately trying to figure out what the derivative, the first derivative of H of X is, our first step that we need to take in order to solve this problem. Prior to performing any sort of differentiation, we need to recall and apply the rules of logarithms for the quotient, product, and power. So we need to go ahead and write that H of X is equal to the natural log of, and if we factor out X2 minus 1, we could write that X. -1 multiplied by X + 1 multiplied by or minus X to the power of 5. All divided by X + 3 to the power of 4, fantastic. And we can also go ahead and write that H of X is equal to the natural log of X minus 1. Plus the natural log of X + 1. Plus 5 multiplied by the natural log of 4 minus X. -4 multiplied by the natural log of X + 3. Fantastic. So, our next step now that we need to take, so our second step. we need to write down the derivative applying the differentiation to each resultant function. So we could go ahead and write that H of X where this prime means, so basically this prime, well it's essentially it's like an apostrophe in the power. This prime symbol means that it's the first derivative of H of X. So H of X is equal to the natural log of. Or rather, the natural log prime of X minus 1. Plus the natural log prime of X + 1. Plus 5 multiplied by the natural log prime of 4 minus X. -4 multiplied by the natural log prime of X + 3. Fantastic. So now our third step that we need to take is we need to recall and use the table of basic derivatives, and we also need to recall and note that the natural log of X is equal to 1 divided by X. And we need to note and recall for this particular problem and some problems that are similar to it. We will have to apply the chain rule for each logarithm, since there are 4 functions, there will be all composite functions. So we're going to be solving for 4 separate answers. So to make things simple, we're going to break them up into 4 separate. Composite functions and and and solve for them separately. So we're solving for four separate answers right now. So that's what we're about to do. So let's start for solving for the natural log of X minus 1. So, for this, this will equal D by D of X minus 1. of the natural log of X minus 1 multiplied by D by DX of X minus 1, which is equal to 1 divided by X minus 1 multiplied by D by DX of X minus 1, which will be all equal to 1 divided by X minus 1. So that is our first composite answer. So now we can solve for natural log of X + 1, and this will be equal to D by D X + 1. So D by D so it's D by D of X + 1, multiplied by the natural log of X + 1, fantastic. And don't forget your parenthesis, and we need to multiply this by D by DX of X + 1, which is a B equal to 1 divided by X + 1, multiplied by D by DX of X + 1, which will be all equal to 1 divided by X + 1. Fantastic. Now we could focus our attention on natural log of the natural log of 4 minus X, and this will be equal to D by D of 4 minus X multiplied by the natural log of 4 minus X. Multiplied by D by DX where D by DX means we're taking the derivative focusing on X, we're taking the derivative of X. Multiplied sorry, so it's D by DX of 4 minus X, which is equal to 1 divided by 4 minus X, multiplied by D by DX of 4 minus X. Which will be equal to, so 4 minus X, and it will be all equal to drumroll, negative 1 divided by 4 minus X, or it will also be equal to 1 divided by X minus 4. Fantastic. So now, we could solve for our last composite function, which is the natural log prime of X + 3, and this will be equal to D by D of X + 3. Multiplied by the natural log of X + 3. Fantastic, multiplied by D by DX of X + 3. Which will be equal to 1 divided by X + 3 and 1 divided by X + 3, so 1 divided by X + 3, multiplied by D by DX of X + 3 will give us the following answer of drum roll, 1 divided by X + 3. Fantastic. So let's put a box around our for final answers for our composite function. So these are not our final answers, these are the composite answers that we need to use. In order to help us solve for our final answer, which we're trying to, once again, we're trying to figure out the first derivative of HF X. So we're going to need these four answers to help us solve for our final answer. So moving right along here. So now, our next step, so step 4, We need to substitute all of our results from step 3 into our expression that we found in step 2, and when we do that, we will find that H of X is going to be equal to 1 divided by X minus 1 plus 1 divided by X + 1 + 5 divided by X minus 4. -4 divided by X + 3. Fantastic. So now our next step, which is our fifth and final step, is we need to note that in step 4 we have already obtained the derivative. Our goal often involves rewriting the result of one fraction just as given in the original function. So we're trying to rewrite the original result of one fraction. Just as given in the original function. So as we should recall a note, the common denominator would involve a product of 4 unique linear functions, and since we would spend more time on algebraic in locations than on differentiation, we may be risking to obtain an error, and we could just leave this as our final derivative. We will only use the original expression X2 minus 1 from H of X to combine the first two terms, so meaning, We can go ahead and write that H. Of X is equal to X + 1 divided by X minus 1 multiplied by X + 1. Awesome. And then we need to add X minus 1 divided by X minus 1 multiplied by X + 1. Awesome. And then we need to add 5 divided by X minus 4, minus 4 divided by X + 3. So that will mean when we simplify, we'll get that H of X. So H of X is equal to 2 X divided by X2 minus 1 + 5 divided by X minus 4. Minus 4 divided by X + 3, and that will be our final answer for the first derivative. Hooray, we did it. So in conclusion, the derivative of the function h X will be H X is equal to 2 x divided by X2 minus 1 + 5 divided by X minus 4 minus 4 divided by X + 3, and that is our final answer. Hooray, we did it. 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 - 1)(x + 2)³ / (1 - 4x)²

Key Concepts

Derivative

Logarithmic Properties

Chain Rule

Watch next