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

Question

Calculate the derivative of the following functions.

y = (1 + 2 tan u)^4.5

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of the function Y E equals 4 + 71 M raised to the core of 8.5. A says it's 52.5 6 squared M multiplied by 4 + 7 tom to 6.5. B 59.56 squared M multiplied by 4 + 71 M to the 7.5. C-52.5 coex squared M multiplied by 4 + 7 M to the power of 6.5 and D negative 59.5 coex squared M multiplied by 4 + 7 M raised to 7.5. Now if we look at the function Y, we can tell it's a composite function. To differentiate that composite function, we can use the chain rule. Now what does that say? Well, recall that if we have a function Y that's equal to U N where U is a function of M, then the derivative of Y with respect to M. Would be equal to the derivative of y with respect to u multiplied by the derivative of u with respect to m. Now in this function here, we can tell that U or inner function is equal to 4 + 70 M. So let's differentiate our function Y with respect to you, you with respect to M and find their product to find the derivative of Y with respect to M, OK? Now, here Given you. Alright, let me write that here. Let me write the talk in red. Eh? And let's start with Y actually. Given that Y is going to be equal to U to the power of 8.5, OK, since U equals 4 + 70 M. Then DUDU, the derivative of Y with respect to you, is going to be equal to 8.5. We bring down the power multiplied by you raise to the core of 8.5 minus 1. We subtract the power. So that's 8.5 multiplied by you to the pore of 7.5, and we know that you. Is equal to 4 + 71 m, OK. So it's gonna be all of that multiplied by 7.5. Next, we know that U is equal to 4 + 7 M. So if we differentiate you with respect to M. OK. Uh, the derivative of 4 is 0 because 4 is a constant, and the derivative of the tangent is equal to sex squared. So the derivative of you with respect to M equals 7 62 M. Now that we have DYDU and DUDM, this tells us then that the derivative of Y with respect to M is going to be equal to. The DYDU 8.5 multiplied by 4 + 7 times M raised to the part of 7.5 multiplied by 76 squared M. Now, if we simplify here, 7 multiplied by 8.5 equals 59.5. So the derivative is going to be equal to 59.5, multiplied by 6 grade M multiplied by 4 + 70m raised to the core of 8.5. This is the derivative of Y with respect to m. If we go back to our answer choices, that tells us 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 = (1 + 2 tan u)^4.5

Key Concepts

Derivative

Chain Rule

Power Rule

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