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

Question

Find the derivative of the following functions.

y = x² (1 - In x²)

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. Find the derivative of the function Y equals 3x3 multiplied by 2 minus the natural log of 3 x 3. Awesome. So it appears for this particular problem, we're asked to figure out our final answer that we're ultimately trying to solve for is we're trying to determine the derivative of this function of Y. So we're trying to find the first derivative of the function of Y. So with that said, Our first step that we need to take is we need to recall a note that the function of Y consists of a product of two functions, and because it's a product of two functions, let us state that F of X is equal to 3x the power of 3, and let us also say that G of X is equal to 2 minus the natural log of 3x the power of 3. And for our second step, let us recall and use the product rule and apply it to this particular problem. So as you should recall a note from the product rule, we need to state that Y. Where this prime symbol indicates the first derivative, and it looks like an apostrophe in the power. So Y is equal to F multiplied by G, which is equal to F multiplied by G, plus F multiplied by G. Fantastic. So now, our 3rd step is we now need to evaluate the two derivatives. So that's our 3rd step. So let's start with F subscript X, so FX. So FX. is equal to 3 x 3, which is equal to 3 multiplied by x 3, which is equal to 3 multiplied by 3 x 2. Which is equal to 9 X squared. And now we could solve for G subscript X or GX and GX is equal to 2 minus the natural log of 3 X to the power of 3. Fantastic. So now, we need to say that this is equal to 2 minus the natural log of 3 X to the power of 3, which is equal to -1 divided by 3x the power of 3 multiplied by 3 X to the power of 3. Which is gonna be equal to -9X2 divided by 3 X 3, which is equal to -3 divided by X, so -3 divided by X is our final answer. And we also need to note that this equation for G, or rather this this function of G, is a composite function and because G is a composite function, we need to recall and use the chain rule. So our 4th step now is we need to substitute the results that we found from our expressions into our step 2 expression. So we need to take our results that we found together and we're taking those results and we're plugging it into the expression from step 2. And when we do that, we will find that Y. is equal to 9 x 2 multiplied by 2 minus the natural log of 3 x 3 plus 3 x 3 multiplied by -3 divided by X. Fantastic. So now, our next step. That we need to take, which is our 5th step. So step 5, is we need to recall and apply the distributive property and multiply, and then we will need to combine like terms, so we need to take Y. is equal to 18 X2 minus 9X2 multiplied by the natural log of 3 X to the power of 3. Awesome, and then we need to take this and we need to subtract 9X to the power of 2. Fantastic. So, when we do a little bit of simplifying, we will find that Y is equal to 9 X2 minus 9 x 2 multiplied by the natural log of 3 x 3. So now our last step, our 6th and final step is we need to factor our expression. So when we do that, we will find that Y is equal to 9 x2. So 9 x 2 or 9 x2 multiplied by 1 minus the natural log of 3 X to the power of 3. And that is our final answer. Hooray, we did it. We saw for final answer, which is the first derivative of the function of Y. So, in conclusion, The derivative of the function of Y is Y is equal to 9 x2 multiplied by 1 minus the natural log of 3x the power of 3. Thank you so much for watching. Hopefully that helped, and I can't wait to see you on the next video. Bye.

Find the derivative of the following functions.
y = x² (1 - In x²)

Key Concepts

Derivative

Product Rule

Natural Logarithm

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