Evaluate and simplify y'.
y = (x²+1)³ / (x⁴+7)⁸(2x+...

Question

Evaluate and simplify y'.

y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷

Accepted Answer

Welcome back everyone. Calculate the derivative of the function y equals 2 x 2 minus 35 divided by x 6 + 5 of 7 multiplied by 4 x + 1 to the power of 6. For this problem, to make it easier for us to differentiate, we're going to apply logarithmic differentiation. So we're going to use the given equation and take the natural log of both sides. We are not changing the equation, but the rules of logarithms will allow us to identify the derivative much more easily. So what we're doing is taking the natural log of y, and on the right hand side we are simply taking the natural log off the given function. So we are going to rewrite it. We have LN of Y equals LN of the top term divided by the bottom term. So the bottom term is X to the power of 6 + 5 is the power of 7, multiplied by 4 X + 1. Raise the power of 6 and from here we're going to apply the rules of logarithms. So on the left hand side we simply have a line of y. And on the right hand side, we're going to apply the power rule and the quotient rule. So first of all, we have to take LN. Of the top function because we have a quotient, so this gives us a line of 2 x 2 minus 3 to the power of fifth, but we have to recall that if we have an exponent we can bring it down in front of the log, right? And then because we then have a denominator we can subtract the terms in the denominator. One of them is our first factor, so that's LN ton of X to the power of 6 + 5, all raises to the power of 7th, and this means that we can write the exponent in front and then we are subtracting the final term in the denominator. Which gives us LM. Of 4 x + 1 raises the power of 6. So once again we can. Bringing down the exponent and this gives us a coefficient. And from here we're going to differentiate both sides. We have to take the derivative of the left hand side, and we have to recall that the derivative of LN of X is equal to 1 divided by X. So in this case we're going to get 1 divided by Y, but Y is a function of X, so we have to multiply by Y based on the chain rule. And this is how we express the derivative on the left. What about the what about the right hand side? Well, we have 5, we're going to factor it out. Then we're differentiating the natural log, so we take 1 divided by the given function which is 2 x cubed minus 3. And now based on the chain rule, we are multiplying that by the derivative of 2 X2, right? So let's make sure that we include the correct exponent 2 X2d. Minus 3. So this is how the chain rule works, right, because we don't have a line of X, we have a line of a function, and we continue with this rule. Then we take minus 7 multiplied by 1 divided by x to the power of 6 + 5 multiplied by. The derivative of X6 + 5. And finally we take -6. Multiplied by 1 divided by 4 x plus 1, that's the derivative of the next slog. And then based on the chain rule, we have to multiply by the derivative of 4 x + 1. Now let's simplify and let's multiply both sides by Y because our goal is to solve for Y. So we get Y equals Y. Is Let's understand that the derivative of 2 x 2 minus 3 is 4 x based on the power rule. Then the derivative of X6 + 5 is 6 X. So the power of fifth based on the power rule, and the derivative of 4 x + 1 is 4 based on the power rule. And of course the derivatives of each constant are 0. So overall, we get 5 multiplied by 4 Xs. This gives us 20 X. And if we look at the exponent, that's 4 x squad, right? The derivative, I'm sorry, that's 4 X, right? The derivative of 2 x 2 is 4 X. So 4 X multiplied by 5 gives us 20 x in the numerator. In the denominator we have 2 X2 minus 3. For the next term, we have 7 multiplied by 6 X5, which gives us 42 x 5. Divided by X to the power of 6 + 5, and for the final term we end up with -6 multiplied by 4, that's -24, divided by 4 X + 1. So we are nearly done. We're going to rewrite it as Y equals. And now what we want to do is simply substitute for Y because we know that the original function Y is 2x2 minus 3 to the power of 5th. And then in the denominator we have X to the power of 6 + 5 of 7, multiplied by 4 X + 1 raises to the power of 6, and all of that is multiplied by. The three functions that we have inside of Prencies. And that's our final answer, so let's label it and conclude the video. Thank you for watching.

Evaluate and simplify y'.
y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷

Key Concepts

Differentiation

Quotient Rule

Chain Rule

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