Find the derivative of the following functions.
y = In √...

Question

Find the derivative of the following functions.

y = In √x⁴+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 the natural log of the cube root of x 6 + 3 x 3. Awesome. So it appears for this particular problem, we're asked to figure out what the derivative of this function of Y is. So now that we know that we're ultimately trying to figure out what the derivative of this function of Y is, Our first step that we need to take in order to solve this problem prior to any differentiation is we need to recall and apply the rules of logarithms to help us simplify the function and let us notice that when we simplify using our rules of logarithms that we should recall, we could go ahead and write that Y is equal to the natural log of. X to the power of 6 + 3 X to the power of 3. All to the power of 1/3. Fantastic. So now, our next step is we need to note and recall that we can manipulate the expression from here, knowing that, and let's make a note, knowing that log base A of B to the power of X is equal to X, multiplied by log base A. Of B So, using this expression, this rule for logs, we could go ahead and give, so we could go ahead and take our function of Y and write it in the following form. We can go ahead and write that Y is equal to 1/3 multiplied by the natural log of X to the power of 6 + 3 x 3. So now, our second step that we need to take is we need to recall and apply the chain rule. And let us note that the inner function U of X will be equal to x 6 + 3 x 3. And the outer function Y will be equal to 1/3 multiplied by the natural log of U. Fantastic. So Y equals 1/3 multiplied by the natural log of U. So according to the chain rule, we could state that Y, where this apostrophe in the power symbol denotes prime, and this states that this will be the first derivative of the function of Y. So Y is equal to 1/3, multiplied by the natural log of u. Prime subscript U multiplied by UX. Which is equal to 1/3 multiplied by 1 divided by U, multiplied by X to the power of 6, plus 3 x 3. To the power of prime or all to prime. So, now our next step is we need to take So we're taking for step 3, we need to substitute, we need to take our value of you, and we're taking U equals X to the power of 6 plus 3 x 3, and we're substituting this value of you into our expression that we found in step 2. So when we do that, we will find that Y is equal to 1 divided by 3 multiplied by X to the power of 6 + 3 X to the power of 3. All multiplied by X to the power of 6 plus 3 x 3. Fantastic. So now, What we need to do for our 4th step. we need to evaluate the remaining derivative, so the remaining of our derivative, and use the power rule, which, as we should recall, the power rule states that X to the power of N is equal to N multiplied by X to the power of N minus 1. Fantastic. So when we do that, we will find that Y is equal to 1 divided by 3 multiplied by x 6 plus 3 x 3. Multiplied by 6 X to the power of 5 plus 9 X to the power of 2. And for our 5th and final step, we need to perform a complete factorization in order to simplify and solve our final result. So we need to take Y equals 3x2 multiplied by 2 X to the power of 3 + 3. All divided by 3 X to the power of 3 multiplied by X to the power of 3 + 3. And when we perform the simplification, we will find that it our final answer. Or Y will be equal to 2 X to the power of 3 + 3 divided by X So X multiplied by X to the power of 3 + 3. Fantastic and Let us take a moment here to recap here. So we can say without a doubt, in conclusion for the derivative of the function of Y, Y will be equal to 2 X 3 + 3, all divided by X multiplied by. X to the 3 + 3, that is our final answer. And we also need to take a moment here to note that, and this is very important, let's make a note of this off to the side in red, that X cannot equal 0 in this particular case, and that's it. That is our final answer. Hooray, we did it. We found our final answer for the derivative of Y. 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 = In √x⁴+x²

Key Concepts

Derivative

Natural Logarithm

Chain Rule

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