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

Question

Calculate the derivative of the following functions.

y = e^2x(2x-7)⁵

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of the function Y equals E 9X multiplied by 7 X minus 19 raised to the power of 10. A says it's E 9X multiplied by 7 X minus 19 to 10 minus 10 E 9X multiplied by 7 X minus 19 to 9. B says it's 9 E 9X multiplied by 7 X minus 19 to 10th, plus 70 E 9X multiplied by 7 X minus 19 to the 9th. C says it's 99 X multiplied by 7 X minus 1910 plus 10 9 X multiplied by 7 X minus 19 to 9th and the D says it's E9X multiplied by 7 X minus 19 to 10th, plus 10e 9X multiplied by 7 X minus 19 to the 9th. Now, when we look at our function, we can tell that it is a product. We're multiplying two terms. So we can apply the product rule of differentiation. Recall that if Y is equal to UV, where U and V are functions of X, then the derivative of Y with respect to X is going to be equal to U DVD X. Plus VDUDX. In other words, we differentiate both terms in our function for Y and then multiply them by the opposite derivatives respectively, and then sum them together. So, let's set both of our terms, OK? In other words, let's let you be equal to E 9X and let V be equal to 7 X minus 1910. Then let's see if we can differentiate them to help us eventually solve for the derivative of y with respect to X. Let's start with you. Now if you equals e to the power of 9 X, then DU X is going to be equal to the derivative of what's in our power. OK. In this case, the derivative of 9 X is 9, multiplied by the original function for E. So that's 9 multiplied by e to the power of 9 X. Next, if we're talking about V, notice that V is a composite function, so we'll have to apply another rule of differentiation, the chain rule to help us find the derivative, OK? Now, given V. Yeah. Uh, being equal to a composite function, OK. So let's say that it's the function of U Z X, OK. Then the derivative of V with respect to X is going to be equal to the derivative of u. Multiplied by the multiplied by our original function for Z, so that's Z of X, multiplied by the derivative of the function Z, eh? In other words here. We're going to differentiate V. Or rather, we're going to differentiate, sorry, we're going to differentiate the alter function for V. In other words, we're gonna bring down the power subtract one from the power, and then we're going to multiply it by the derivative of the inner function of V, which is 7 X minus 19. So since we equals 7 X minus 19 raise to the power of 10, then DVDX is going to be equal to, we're gonna bring down the power of 10, so that's 10, multiplied by ZX. OK, original function with the power subtracted by 1, so that's 7 X minus 19, sorry, raise to the power of 9. Multiplied by the derivative of 7 X minus 19. Now, to find that, we can find the derivative of 7 X which is 7, and the derivative of negative 19, which is 0. So all of this is being multiplied by 7. And then we can rewrite this as 10 multiplied by 7 or 70 multiplied by 7 X minus 19 raised to the power of 9. Now that we have DVDX. OK. And we have DUDX. We can apply the chain rule. Because no, by the chain rule, sorry, the product tool, the product rule. No, by the product tool DUIDX is going to be equal to. E to the power of 9 X. OK. Multiplied by DDX which is 70, multiplied by 7 X minus 19, raise to the power of 9. Plus V, which is 7 X minus 19, raises to the power of 10 multiplied by DUDX which we found to be 9 E to the power of 9 X. Now we can multiply E to the 9 X by 70 and write our term race to the 10th first. And when we do that, we're going to get 9 E to the power of 9X multiplied by 7 X minus 19 to 10th, OK. Plus 70 e to the 9 x multiplied by 7 X minus 19 to the 9th. Thus this is the derivative of y with respect to X. If we look back on our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Calculate the derivative of the following functions.
y = e^2x(2x-7)⁵

Key Concepts

Derivative

Product Rule

Chain Rule

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