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

Question

Calculate the derivative of the following functions.

y = ⁴√(2x / (4x - 3))

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of the function Y equals the 4th root of 2 X divided by 6X minus 1. A says it's 1 divided by X multiplied by the 4th root of 8 X cubed multiplied by 6 x minus 1. B says it's the negative value of 1 divided by 2 multiplied by the absolute value of 6 X minus 1 multiplied by the 4th root of 8XQed multiplied by 6 x minus 1. C says it's 2 divided by X multiplied by the 4th root of X cubed multiplied by 6 x minus 1 squared. And the D says it's the negative value of 2 divided by X multiplied by the 4th root of X cubed multiplied by 6 X minus 1. Now if we are going to differentiate the function to find the derivative of Y, if we take a good look at Y, we can tell that it is a composite function. So we'll need to use the chain rule to differentiate Y. And then when we look at the term inside the composite function, we can tell it's a quotient. So we'll need the quotient rule. Now what do both of those rules say? Well, let's start with the chain rule, OK? Recall that by the chain rule. OK. By the chain rule. If Y is equal to? A function you raised to some power N, OK. Then DUIDX is going to be equal to DUIDU divided by, sorry, DUIDU multiplied by DUDX, OK? I know in this case Then if we apply the chain rule, we can let you be equal to 2 X divided by 6 X minus 1, OK. Which then makes why if we do that then Y is going to be equal to the 4th root of you or you to the pore of 1/4. Now with that idea in mind then. We can say that D YDU is going to be equal to 1/4 of you we bring down the power, so that's 14th multiplied by U raised to the core of 1/4 minus 1, which als -3/4, and we could rewrite this even further to be 1 divided by 4 you raised to the core of 3/4, OK. So now we've found the DYDU. Next, we need to figure out DUD X. In other words, we're differentiating U or term here 2 X divided by 6 X minus 1 with respect to X. And like we said earlier, we can use the quotient rule. I know by the casher rule, OK. That basically means. That is. We have a function in this case you. Being equal to V divided by Z. Then DUDX, the derivative of you with respect to X, is going to be equal to Z DVDX minus VD ZDX divided by Z2d. In other words, we are going to differentiate our or numerator and denominator and then plug them into the formula for the quotient rule, OK? So now, if we let. V be equal to 2 X, OK. And we let. Z be equal to the denominator 6 X minus 1, OK. Then DVDX will be equal to 2. D Z D X will be equal to 6 and thus this implies then that DUD X. Is going to be equal to 6 X minus 1. OK. Multiplied by 2. Minus 6 multiplied by 2 X based on a quotient rule all divided by 6 X minus 1 or squared. And if we simplify what's going on in our numerator, 2 multiplied by 6 X minus 1 equals 12 x minus 2, OK. I know if we take it a step further, 12 x minus 12 x equals 0, so we're left with negative 2 divided by 6 X minus 1 squared for DUDX. Now if we go back to our chain rule here, we're going to multiply DUIDU by our newly found value for DUDX. So now this implies then. Dot DUIDX, OK. Is going to be equal to 1 divided by 4 u. We already know that U is 2 x divided by 6 X minus 1 raised to the power of 3/4 multiplied by -2 divided by 6 X minus 1 squared. Now let's try to rewrite our expression and simplify. Now, here, 2 goes into for 2 times, OK. So now we're left with a negative value of 1, OK, divided by 2 multiplied by 2 X to the power of 3/4, OK. Multiplied by 6 X minus 1 raised to the core of 5/4. Because remember if, sorry, 6 X minus 1 would be raised to 3/4 and if we're multiplying it by 6 X minus 1 squared since they both have the same base, we would add the powers, OK? So now, OK, in this case. We think about it here. What that really means. So what that would mean it would be 6 X minus 1 to the point of negative 3/4 + 2, which is how we ended up with 5/4, OK. Now here we could also write all our terms in the denominator as powers of fours. Because if we think about it, OK, we could rewrite this. OK. As 1 divided by 2 to 4/4 because 2 is being raised to the pore of 1, OK, multiplied by 2 of 3/4, multiplied by X to the power of 3/4. Multiplied by 6 X minus 1 to the power of 5/4, OK. Now when we do that here. Then we can say that this is going to be equal to the negative value of 1 divided by 2 multiplied by the absolute value of 6 x minus 1 multiplied by the 4th root of 2 cubed. X cubed multiplied by 6 X minus 1. And now, if we rewrite 2 cubed as it, then this is going to be equal to the value of -1 divided by 2 multiplied by the absolute value of 6 X minus 1 multiplied by the 4th root. Of 8 excued multiplied by 6 X minus 1. Thus this is going to be the derivative of Y with respect to X. Now if we look back at our answer choices, that tells us then that B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Calculate the derivative of the following functions.
y = ⁴√(2x / (4x - 3))

Key Concepts

Derivative

Chain Rule

Quotient Rule

Watch next