Computing the derivative of f(x) = e^-x
c. Use part...

Question

Computing the derivative of f(x) = e^-x

c. Use parts (a) and (b) to find the derivative of f(x) = e^-x.

Accepted Answer

Below there, today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the derivative of F of X is equal to the power of -12X using the following information. D by DX of the power of -12 X is equal to the power of -12 x multiplied by the limit evaluated at H approaches 0 of the power of -12 minus 1 divided by H. And the limit valuated at H approaches 0 of E to the power of H minus 1 divided by H is equal to 1. Awesome. So it appears for this particular problem, we're asked to figure out what the derivative of FX is using the following information provided to us by the problem itself. So with that said, let's read off our multiple choice answers to see what our final answer for the derivative might be. A is 12 multiplied by the power of -6 X. B is -12 multiplied by the power of -12 X. C is 12 multiplied by E to the power of -12 X. And finally, D is 24 multiplied by E to the power of 12 X. Awesome. So first off, we need to Let the value W equal -12 H, and we also need to note that then we could state that as H approaches 0, we can also say that W approaches 0. So with that said, we could go ahead and write that the limit evaluated at H approaches 0 of E to the power of -12 H. 1 divided by H is equal to the limit evaluated at W approaches 0 of E to the power of minus 1 divided by negative W divided by 12. Which is all equal to the limit evaluated at W approaches 0 of -12 multiplied by E to the power of W minus 1 divided by W. Which is all equal to -12 multiplied by the limit, which is evaluated at W approaches 0 of E W minus 1 divided by W. And since it is given that the and let's note, let's make a little note of that note of this fact, so let us know that it's given that the limit as H approaches 0 of E to the power of H minus 1 divided by H is equal to 1, this will mean that as a result that this will follow that the limit. As W approaches 0 of E to the W minus 1 divided by W is also equal to 1. So, with that said, we need to substitute in the value of 1 for the limit evaluated at W approaches 0 of E W minus 1 divided by W. So when we do that, we will find that the limit, which is evaluated at H approaches 0 of E to the power of -12 H minus 1 divided by H. Is all equal to -12 multiplied by the limit evaluated at a W approaches 0 of E W minus 1 divided by W, which is equal to -12 multiplied by 1, which is equal to -12. So therefore, using the first equation that we established, we can state therefore, that we could write that D by DX of E to the power of -12 X is equal toe the power of -12X multiplied by the limit evaluated at H approaches 0 of E to the power of -12 H. -1 divided by age. Fantastic, and this will be equal to the power of -12 X multiplied by ne12, which will mean that it's all equal to -12 multiplied by E to the power of -12 X, and that is our final answer for the derivative. Hooray, we did it. And this corresponds to multiple choice answer letter B, which once again is -12 multiplied by E to the power of negative 12 X. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Computing the derivative of f(x) = e^-x
c. Use parts (a) and (b) to find the derivative of f(x) = e^-x.

Key Concepts

Derivative

Exponential Functions

Chain Rule

Watch next