Derivatives Find and simplify the derivative of the following ...

Question

Derivatives Find and simplify the derivative of the following functions.

h(w) = w⁵/³ / w⁵/³+1

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the derivative of H X equals X 7/4 divided by X 7/4 + 5. A says it's X 3/4 divided by 4 multiplied by X 7/4 + 5 squared. B 7 X 3/4 divided by 4 multiplied by X 7/4 + 5 squared. C 35 x 3/4 divided by X 7/4 + 5 squared, and the D 35 X to the 3/4 divided by 4 multiplied by X 7/4 plus 5 squared. Now how are we going to differentiate H of X? What do we know? Well, notice that H of X is a cautient and to differentiate a quotient, we can use the quotient rule. By the quotient rule of differentiation. It basically tells us that if we have a term Y E equals U divided by V, OK, where U and V are functions of X, then the derivative of Y with respect to X is going to be equal to Vu minus UV all divided by V2. So in other words, we're going to differentiate our numerator and denominator and substitute them into our quotient rule to find the derivative. So let's go ahead and do that. Notice our numerator is 74 X 7/4, that's U, and our denominator V is X 7/4 + 5. When we let you be equal to X 7/4, then we can differentiate you using the power rule. That means we're gonna bring down our power 7/4 and then subtract one from the original power in our term. So that's x 7/4 minus 1, which would leave us with 3/4. Likewise, if we let V be equal to X 7/4 plus 5, OK, then the derivative of V is going to be equal to 7/4 multiplied by X 3/4. Again, we can apply thepo rule, and then when we differentiate 5, it leaves us with 0. The derivative of a constant is 0. Thus, the derivative of U and V are both 7/4 of X to the 3/4. Know that we have the derivatives, we can apply the quotient rule. Because now this tells us then that the derivative of H of X will be equal to V. X to the 7/4 + 5 multiplied by the derivative of U 7/4 of X to the 3/4, OK, minus U X 7/4. Multiplied by V 7/4 of X to the 3/4. And all of that is being divided, OK, by the D V, OK, squared. So that's X to the 7/4 plus 5. Square Now we can try to simplify our expression. Notice here that both of them have the term 7/4 of X 3/4s in common. So if we factor that out, OK. Then we're left with X 7/4 + 5 minus X 7/4 as a terminal numerator. Added by our denominator. And now we can factor out X sorry, we can subtract X 74 from X to the 7/4 and now it will be multiplying 5 and 5 multiplied by 7/4 equals 35/4, OK? So, no, it's gonna be 35/4 of X to the 3 divided by 4, and we could write 35 in our numerator, OK, write back X to the 3/4, and in our denominator, we can write 4, OK? So, that's 4. We write back our original X to the 7/4 plus 5 square. Thus, this is going to be the derivative of H of X. If we look back at our answer twice, that means D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Derivatives Find and simplify the derivative of the following functions.
h(w) = w⁵/³ / w⁵/³+1

Key Concepts

Derivatives

Quotient Rule

Simplification of Derivatives

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