Find the derivative of the following functions.
y = x² I...

Question

Find the derivative of the following functions.

y = x² In x

Accepted Answer

Be 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. Find the derivative of the function Y equals 3 x 3 multiplied by the natural log of X. Awesome. So it appears for this particular prong we're trying to figure out the derivative of this function of Y. So we're trying to find the first derivative of this function of Y. So with that said, our first step in order to solve this problem is we need to analyze our given function. And factor out the constant. So when we do that, we will find that Y is equal to 3, multiplied by X to the power of 3, multiplied by the natural log of X. Which you could put parentheses around the X from the natural log of X, or you could just write the natural log of X. It's whatever you prefer. So now we need to recall and note that the part inside of the parenthesis consists of a product of two functions, which are F equals X the power of 3 and G equals the natural log of X. So for this particular differentiation, we need to recall and apply the product rule, meaning we need to take Y, which this prime symbol indicates the first derivative of this function of Y, and this prime symbol looks like an apostrophe in the power. So, so we could write that Y is equal to 3 multiplied by F multiplied by G, which is equal to 3 multiplied by F. multiplied by G plus F multiplied by G. Fantastic. So now, our second step that we need to take is we need to evaluate the derivatives of F and G with respect to X, and for F, we need to recall and apply the power rule of differentiation, and for G, we may use the table of basic derivatives. So let us solve for F of X first. So F of X will be equal to D by DX, which this is a fancy way of saying we're taking the derivative with respect to X. So D by DX of X to the power of 3 is equal to 3 x to the power of 3 minus 1, which will be equal to 3 x 2. And similarly for G of X, so G of X will be equal to D by DX of the natural log of X, which will be equal to 1 divided by X. Now we can move on to our third step. Our third step is we need to substitute the derivatives into the expression that we found in step 1, and we will also need to substitute the original functions into our given product rule. And when we do that, we will obtain the following results. We can write that Y. is equal to 3 multiplied by 3 X squad multiplied by the natural log of X plus X to the power of 3 multiplied by 1 divided by X. Fantastic. So now our fourth step is we need to distribute the terms and simplify. So when we do that, we could say that Y is equal to 9X2 multiplied by the natural log of X plus 3 X 3 divided by X. Which we could also simplify a little bit more and write that Y is equal to 9 X 2 multiplied by the natural log of X plus 3X to the power of 2. Fantastic. So now our fifth and final step is we need to factor out the greatest common factor, which in this case is 3x the power of 2. So when we do that, we will find that Y is equal to 3 x 2 multiplied by 3 multiplied by the natural log of X + 1. Fantastic, and that is our final answer. Hooray, we did it. So therefore, without a doubt, the derivative of the function of Y, so the derivative of the function of Y will be Y is equal to 3x2 multiplied by 3, multiplied by the natural log of X plus 1. 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 = x² In x

Key Concepts

Derivative

Product Rule

Natural Logarithm

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