Calculate the derivative of the following functions.
y =...

Question

Calculate the derivative of the following functions.

y = √x+√x+√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 the function below. F of X is equal to the cube root of X plus the cube root of X plus the cube root of X. Awesome. So it appears we have a pretty big function here with a lot of cube roots, but ultimately what we're trying to do is we're trying to take the derivative. Of this specific function. So we're trying to find the derivative of F of X. So with that in mind, our first step in order to solve for this derivative is we need to recall and apply the chain rule of derivatives, which, as we should recall, we need to note that D by DX. Of you of V of X. is equal to U U V X multiplied by VX. Awesome. So, with that said, we need to take F of X. So now we need to apply the chain rule to our specific function. So we need to take F of X. is equal to. One divided by. 3 multiplied by X plus the cube root. Of X plus the cube root of X. Awesome. So, with that said, and note that there's a cube root inside of a cube root, that's important to note. Awesome, and we need to make sure that this, so X plus the cube root of X plus the cube root of X, is to the power of 2/3, and then we need to take and multiply this by D by DX of X plus the cube root of X plus the cube root of X. Fantastic, and this will be equal to. 1 divided by 3 multiplied by X plus the cube root of X plus the cube root of X. And don't forget that this is to the power of 2/3, and then this will be multiplied by 1 plus 1 divided by 3. Multiplied by X plus the cube root of X to the power of 2/3 multiplied by D by DX of X plus the cube root of X. Fantastic. And then this will be equal to 1 divided by 3 multiplied by X plus the cube root of X plus the cube root of X. And this once again is to the power of 2/3, and this will be multiplied by 1 + 1 divided by 3 multiplied by X plus the cube root of X to the power of 2/3. And then this will be multiplied by. 1 + 1 divided by 3 X to the power of 2/3. Fantastic. And don't forget your parenthesis. Awesome. So we have a little bit of a lengthy problem here, but don't worry, we're almost done. So take a breath. So now, our next step is we need to set this is gonna be equal to. 1 divided by 3 multiplied by X plus the cube root of X plus the cube root of X, and don't forget, once again, this is to the power of 2/3. Fantastic. And then we need to multiply this by 1 + 1 divided by 3 multiplied by X plus the cube root of X, and this is to the power of 2/3. Then we need to multiply this by 3 X to the power of 2/3 plus 1 divided by. 3 X to the power of 2/3. Fantastic. And this will be equal to one divided by. 3 multiplied by X plus the cube root of X plus the cube root of X to the power of 2/3, fantastic, and then this will be multiplied by 1 plus. 3 X to the power of 2/3 plus 1, all divided by 9 X to the power of 2/3. Multiplied by X plus the cube root of X to the power of 2/3. Fantastic. And finally. So now, our next step in which We need to do a little bit more simplifying, so bear with me. So, we need to simplify a little bit more. So, we need to keep, we need to keep going. We're almost done, but we need to keep going. So we need to take one divided by 3 multiplied by X plus the cube root of X plus the cube root of X to the power of 2/3, fantastic. And then this is multiplied by 9 X to the power of 2/3. Multiplied by X plus the cube root of X. To the power of 2/3 + 3 x to the power of 2/3 plus 1. All divided by 9 X to the power of 2/3, multiplied by X plus the cube root of X to the power of 2/3. Fantastic. And finally, doing one last run, we will get that this is equal to. 9 X to the power of 2/3, so 9 x 2/3 multiplied by X plus the cube root of X to the power of 2/3. Plus 3 x to the power of 2/3 plus 1. All divided by 27 X to the power of 2/3 multiplied by X plus the cube root of X to the power of 2/3 multiplied by X plus the cube root of X plus the cube root of X. And this is all and then this X plus the cube root of X2 plus the cube root of X is to the power of 2/3. And that's it. That is our final answer. Hooray, we did it. We saw for our final derivative. Hooray! 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.
y = √x+√x+√x

Key Concepts

Derivative

Chain Rule

Power Rule

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