If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:3:40

AP.CALC:

FUN‑3 (EU)

, FUN‑3.C (LO)

, FUN‑3.C.1 (EK)

let's say that Y is equal to 7/2 the x squared minus X power what is the derivative of Y derivative of Y with respect to X and like always pause this video and see if you can figure it out well based on how it this has been color-coded ahead of time you might immediately recognize that this is a composite function or it could be viewed as a composite function if you had a V of X which if you had a function V of X which is equal to 7 to the X power and you had another function U of X U of X which is equal to x squared minus X then what we have right over here y Y is equal to 7 to something so it's equal to V of and it's not just V of X it's V of U of X instead of an X here you have the whole function U of X x squared minus X so it's V of U of X and the chain rule tells us that the derivative of Y with respect to X and you'll see different notations here sometimes you'll see it written as the derivative of V with respect to u so V prime of U of x times the derivative of U with respect to X so that's one way you could do it or you could say that this is equal to this is equal to the derivative the derivative of V with respect to X derivative or sorry derivative of V with respect to u DV d u times the derivative of U with respect to X derivative of U with respect to X and so either way we can apply that right over here so what's the derivative of V with respect to u what is V prime of U of X well we know we know let me actually write it right over here if V of X is equal to 7 to the X power V prime of X would be equal to and we prove this in other videos where we take x derivatives Exponential's of bases other than e this is going to be the log of 7 times 7 to the X power so if we are taking v prime of U of X then notice instead of an X everywhere we're going to have a U of X everywhere so this right over here this is going to be natural log of 7 times 7 to the instead of saying 7 to the X power remember we're taking V prime of U of X so it's going to be 7 to the x squared minus x power x squared x squared minus X power and then we want to multiply that times the derivative of U with respect to X so u prime of X well that's going to be 2x to the 1st which is just 2x minus 1 so we're going to multiply this times 2x 2x minus 1 so there you have it that is the derivative of Y with respect X you could we could try to simplify this or I guess reexpress it in different ways but the main thing to realize is look we're just going to take the derivative of the 7 to the this to the U of X power with respect to U of X so we treat the U of X the way that we would have treated an X right over here so it's going to be natural log of 7 times 7 to the U of X power we take that and multiply that times u prime of X and once again this is just an application of the chain rule

AP® is a registered trademark of the College Board, which has not reviewed this resource.