15–48. Derivatives Find the derivative of the following functi...

Question

15–48. Derivatives Find the derivative of the following functions.

y = e^x x^e

Accepted Answer

Welcome back, everyone. For the given function, find the derivative. Y equals e the power of 5x multiplied by x2. So let's understand that in this problem we're given a product of two functions. Let's suppose that F is equal to E to the power of 5 X and G is equal to x2. So we're looking at differentiating the product F multiplied by G, and we are going to use the product rule. It says that the derivative of F multiplied by G is FG plus FG. So we want to identify two derivatives. Let's begin with F. So we are differentiating E to the power of 5 X, which is E to the power of 5 X, right, but we need to apply the chain rule. So we're multiplying it by the derivative of the inner function in this case 5X, and the derivative of 5 X is simply 5, so we're going to get 5 e to the power of 5 X, and that's our F. Now for G we're going to use the power rule because we're differentiating X2. We have to bring down the exponent, that's 2, and we decrease the original exponent by 1 unit, so we get 2 X. And now applying the rule, we can say that the derivative of Y is equal to virtual F 5 E to the power of 5 X multiplied by G which is X2. And then we are adding F which is E to the power of 5 X, multiplied by G which is 2 X. And now we want to simplify the result, right? We can essentially factor out X and we can factor out E to the power of 5 X, which leaves us with 5 X for the first term, and for the second term we get 2. So the final answer is Y equals x multiplied by E to the power of 5 X multiplied by 5 x + 2. Thank you for watching.

15–48. Derivatives Find the derivative of the following functions.y = e^x x^e

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15–48. Derivatives Find the derivative of the following functions.
y = e^x x^e