Evaluate and simplify y'.
y = 4u²+u / 8u+1

Question

Evaluate and simplify y'.

y = 4u²+u / 8u+1

Accepted Answer

Hello. In this video, we are going to be calculating the derivative of the given function. The function given to us in this problem is Y is equal to 2 T cubed plus 3 T squared, divided by 4T + 2. So, in order to take the derivative of this function, notice that the function is given to us as a rational function. In order to take the derivative of a rational function, we will be using the quotient rule. So the quotient rules defined as the following. If we want to take the derivative of a rational function, we will assign the numerator its respective function F of X and the denominator a respective function G of X. Then the quotient rule is defined as G of X multiplied by the derivative of F of X minus F of X multiplied by the derivative of G of X, all divided by G of X squared. So, in order to take this derivative, let's go ahead and find the 4 pieces of information we need. First, we will label our numerator as the F of X function. F of X will be 2 T cubed plus 3 T squared. Next, we will assign the denominator, the GovXx function. G of X will be 4 T + 2. Now we need the derivatives of the individual functions. We will take the derivative of the numerator by using the power rule. That derivative F of X will be 6 T squared. Plus 60. And by the power rule, the derivative of the denominator G of X will just be 4. Now that we have the pieces of information we need, let's go ahead and place this together in the quotient rule to find the derivative of the original function. The quotient rule will give us the derivative Y equal to the denominator, which is 4T + 2 multiplied by the derivative of the numerator, which will be 6T2 + 6T minus the original numerator, which is 2T cubed plus 3 T squared. Multiplied by the derivative of the denominator, which is 4, all divided by the original denominator squared, which will be 4 T + 2 quantity squared. Now what we need to do is we need to simplify the numerator. We will start by foiling the quantities 4 T + 2 and 6 T 2 + 6T together. That is going to give us 24 T cubed plus 24 T squared. Plus 12 t squared. Plus 12 t And -1, multiplied by 4, will give us -4, and distributing -4 into 2 T cubed plus 3 T squad will give us the terms negative 8T cubed minus 12 T squad. And this will be all divided by 4 T + 2 quantity squared. Now we need to combine light terms. We can combine 24 T cubed and negative AT cubed. We can also combine 24 T 2, 12 T2, and negative 12 T squad. This will simplify the overall numerator to be 16 T cubed. Plus 24 T squared. Plus 12 T. And again, this will all be divided by 40. Plus 2 quantity squared. Next, we can further simplify by factoring out a 4 T from the numerator and by factoring out a 4 from the denominator. That will allow us to simplify the function to be 4 T multiplied by the quantity, 4 T squared. Plus 60. Plus 3, all divided by 4 multiplied by the quantity, 2 T plus 1 quantity squared. Finally, if we reduce the multiples of 4 in both the numerator and denominator to 1, that will give us a final derivative Y is equal to T multiplied by 4 T2 + 6 T + 3, all divided by 2 T plus 1 quantity squared. And this is going to be the simplest form of the derivative Y. And with that being said, the solution to this problem is going to be C. So, I hope this video helps you in understanding how to approach this type of derivative, and I will go ahead and see you all in the next video.

Evaluate and simplify y'.y = 4u²+u / 8u+1

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Evaluate and simplify y'.
y = 4u²+u / 8u+1