Find the derivative of the following functions.
y = e^-x...

Question

Find the derivative of the following functions.

y = e^-x sin x

Accepted Answer

Welcome back, everyone. Find the derivative of the function Y equals e to the power of x cosine x. We're given 4 answer choices A says Y equals negative eat the power of x x x. B says Y equals the power of x multiplied by the sum between cosine of X and sin x. C says that Y is equal to. E to the power of X multiplied by the difference between cosine of X and sine X, and D says that Y equals e to the power of x x x. So now let's rewrite the function we're given the Y equals e to the power of x cosine of X. We want to evaluate the derivative Y, which is the derivative of e to the power of X multiplied by cosine of X, and we can notice that this is a product of two functions. Let's assign F equals e to the power of X, the first function. And then G equals cosine x, our second function. So we will have to apply the product rule. It says that if we multiply F by G and if we want to evaluate the derivative, it is going to have a form of F G plus F G. So we have F, we have G. We want to valid F, which is the derivative of eat to the power of X, and based on the table of basic derivatives, this is eat the power of x. And then we want to evaluate G, which is the derivative of cosine of X, and based on the same table, the value is negatives of x. We may also recall this. We have been using this a lot, right? So now we're going to substitute these into the expression. So Y is equal to F which is e to the power of x multiplied by G, which is cosine of x, plus F G. So we're taking E to the power of X multiplied by G. So that's negatives of X. And now we can factor out it's the power of X, right? And then in prints we're going to have cosine of X minus sign of X because we have that negative sign. Which essentially means that the first derivative corresponds to the answer choice C Y equals each the power of x multiplied by the difference between cosine x and sine x. Thank you for watching.

Find the derivative of the following functions.
y = e^-x sin x

Key Concepts

Derivative

Product Rule

Exponential and Trigonometric Functions

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