Second derivatives Find d²y/dx² for the following functions.

Question

y = e^-2x²

Accepted Answer

Welcome back, everyone. Determine the second derivative for the function Y equals e to the power of negative 15 x cubed. If we want to identify the second derivative, let's begin with the first derivative. So we want to find the derivative of E to the power of negative 15 x cubed. Notice that this is a composite function, right? So we have to apply the chain rule. The derivative of E to the power of X is e to the power of X. So first of all, we can say that the derivative of the given function is E to the power of -15 x cubed, but our exponent. Consists of a variable that is not x, so we have to apply the chain rule, and this means that we're multiplying the derivative by the derivative of the inner function, and our function is -15 x cubed. So now we get each to the power of -15 X cubed multiplied by the derivative of negative 15 X cubed, and according to the power rule, this gives us -453 multiplied by -15. And then multiplied by x to the power of 2 because we're decreasing 3 by 1 unit. Now let's simplify. We're going to get -45 x 2 each to the power of -15 x cub. That's our first derivative. Now for the second derivative, we have to differentiate the previous result, and for simplicity we can factor out the constant. We're basically finding the derivative of X2d multiplied by e to the power of negative 15 x cubed. So now we're looking at a product, right? We have two functions, and according to the product rule, we want to take the derivative of the first function. So we're taking the derivative of X2d. Which is then multiplied by the second function eats the power of -15 x cubed, and we are adding the first function multiplied by the derivative of the second function. So x2 multiplied by e to the power of -15 x cubed and specifically its derivative, right? Now, let's evaluate the remaining derivatives, and then we are going to simplify the derivative of X2 is 2 X. Which is multiplied by E to the power of -15 X cubed. And then we are adding. X squad multiplied by the derivative that we already have, right? We already know that the derivative of each the power of negative 15 x cubed is. 45 X 2 e to the power of -15 XQ. We already have all of the derivatives we simply want to. Simplify the answer. Now, let's take. -45 and turn it into 45 and we're just going to swap the signs. Notice that if we take 45, specifically we can eliminate the negative sign from the second term, right? And this is going to give us 45. X2 multiplied by X2 is X the power of 4th. We then get E to the power of. -15 X cubed and then we are subtracting, right? We're subtracting two acts. E to the power of -15 X cubed. For simplicity we can now fax her out. X, right, because between X and X to the 1, X to the 1st has slower exponent, and then we can also factor out E to the power of -15 XQ because it's a common term. And now in pris we end up with 45. X4 becomes X cubed. And we are subtracting too because this is all that remains from the second term. And that's our final simplified 2nd derivative. Thank you for watching.

Second derivatives Find d²y/dx² for the following functions.
y = e^-2x²

Key Concepts

Second Derivative

Exponential Functions

Chain Rule

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