Evaluate and simplify y'.
y = (3x+5)¹⁰ √x²+5 / (x³+...

Question

Evaluate and simplify y'.

y = (3x+5)¹⁰ √x²+5 / (x³+1)⁵⁰

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with derivatives. This says to calculate the derivative of the function Y is equal to the quantity of 7 X plus 9 in quantity rates to the power 15 multiplied by the square root of the quantity of X2 + 8 in quantity, divided by the quantity of X4 plus 5 in quantity rates to the power 40. Now, in order to calculate this derivative, we're going to need to use our quotient rule. So recall for the quotient rule, that if we take the derivative of the quantity of U divided by V. That that is going to be equal to V multiplied by U minus U multiplied by V and that whole quantity is divided by B squad. So here we're going to identify you as our numerator, so that means U is going to be equal to the quantity of 7 X plus 9 in quantity raised to the power 15, multiplied by the square root of the quantity of X2 + 8. Now I need to take a derivative of this function for you, so we'll have U is equal to, and here we will need to use our product rule. So we're going to have the derivative of our first term, which is going to be 15, multiplied by the quantity of 7 X plus 9 in quantity rates to the power 14, multiplied by 7. And then that gets multiplied by our second term, which is the square root of the quantity of X2 + 8. They will have plus our first term, which is the quantity of 7 X plus 9 in quantity rates to the power 15 multiplied by the derivative of our second term, which is going to be the quantity of X divided by square root of the quantity of X2 + 8. So now we need to simplify this expression. This is going to be equal to. 105 Multiplied by the quantity of 7 X plus 9 in quantity raised to the power of 14. Multiplied by the square root of the quantity of X2 + 8 in quantity, and we'll have plus x multiplied by the quantity of 7 X plus 9 in quantity rates of the power 15 divided by the square root of the quantity of X2 + 8. So we can combine these two terms into one term with a common denominator. So this is going to be equal to. 105 multiplied by the quantity of 7 X plus 9 in quantity raised to the power 14 multiplied by the quantity of X2 + 8 in quantity plus x multiplied by the quantity of 7 X plus 9 in quantity raised to the power of 15. And this whole quantity is divided by the square root of the quantity of X2 + 8. Now I can factor out a quantity of 7 X plus 9 in quantity rates to the power 14 out of both terms in the numerator. So this is going to be equal to the quantity of 7 X plus 9 in quantity raised to the power of 14. And then with the terms that are left in the numerator, I'm going to just combine like terms. This is going to give me 112 X squared plus 9 X. Plus 840. In quantity, and that gets divided by The square root of the quantity of X2 + 8. So now we need to look at the, which is going to be our denominator. This is going to be equal to the quantity. Of X to the 4th plus 5 in quantity raised to the power 40. And I'll need to take a derivative of that quantity, so I'll have V is equal to. 40 multiplied by the quantity of X 4th plus 5 in quantity raised to the power 39, multiplied by 4 X cubed. And simplifying that expression, we will get that. B is equal to 160 cubed. Multiplying the quantity of X 4th plus 5 in the quantity raised to the power 39. So, now we can apply our quotient rule. Now, we've calculated all these quantities in order to find um a derivative of Y. So, Y. Is going to be equal to. And here we're going to have our V, which is the quantity. Of X to the 4th plus 5, in quantity rates to the power 40. Multiplied by U, which is the quantity of the quantity of 7 X plus 9 in quantity raised to the power 14, multiplied by the quantity of 112 X2 + 9 X + 840. In quantity, divided by. The square root of the quantity of X2 + 8. In quantity, then we will have minus U, which is The quantity of 7 X plus 9 in quantity raised to the power of 15. Multiplied by the square root of the quantity of X2 + 8. In quantity, and that gets multiplied by V, which is the quantity of 160 ex cubed. Multiplied by the quantity of Next to the 4th, Plus 5 in 20 raised to the power 39. And all of that is divided by the quantity of V2d, which is the quantity of X to the fourth. Less 5 in quantity raised to the power 80. So now we need to simplify this expression. And so we're going to factor out a couple things out of our numerator, so we'll have this being equal to. We'll factor out 39 powers of the quantity of X of the fourth plus 5. So that will be our first term. We'll have X to the the quantity of X to the 4th plus 5, in quantity rates of power 39. We can also factor out a quantity of 7 X + 9 in quantity rates of the power of 14 of both terms in our numerator. So, we'll have this being multiplied by the quantity of 7 X plus 9 in quantity rates of the power of 14. And then, what we'll have left is the quantity of 112 X squared. Plus 9 X Plus 840 in quantity, and that gets multiplied by quantity of X to the fourth plus 5, since we had one quantity left over, when we factored out 39 of the others. Then we will have minus. 160 cubed. Multiplied by the quantity of 7 X plus 9 in quantity. And here we're gonna go ahead and combine everything over a common denominator, so this is going to be multiplied by the quantity of X2 + 8 in quantity. And so this whole quantity is divided by The quantity of. The quantity of X 4th plus 5 in quantity, raise to the power 80. Multiplied by the square root of the quantity of X2 + 8. So now we can cancel out 39 powers of the quantity of X of the fourth plus 5 out of the denominator. So this is going to be equal to the quantity of 7 X plus 9 in quantity raised to the power 14, multiplying the quantity of the quantity of 112 X2 + 9 X. Plus 840 in quantity, multiplied by the quantity of X4 plus 5 in quantity, minus 160 X cubed. Multiplied by the quantity of 7 X plus 9 in quantity, multiplied by the quantity of X2 + 8. In quantity And that is divided by The quantity of 14th. Plus 5, in quantity raised to the power 41, multiplied by the square root of the quantity of X2 + 8. And so this is the answer that we were actually looking for. Or the solution to this problem. So, I hope that this explanation has been useful, and I'll actually see you in the next video.

Evaluate and simplify y'.
y = (3x+5)¹⁰ √x²+5 / (x³+1)⁵⁰

Key Concepts

Differentiation

Product Rule

Quotient Rule

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