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:9:00

AP.CALC:

FUN‑3 (EU)

, FUN‑3.A (LO)

, FUN‑3.A.2 (EK)

we've been given some inch for interesting information here about the functions F G and H for F they tell us for given values of X what f of X is equal to and what F prime of X is equal to then they define G of X for us in terms of this kind of absolute value expression and then it define H of X for us in terms of both f of X and G of X and what we're curious about is what is the derivative what is the derivative with respect to X of h of x of h of x at x is equal to 9 and so i encourage you to pause this video and think about it on your own before I work through it so let's think about it a little bit so another another way just to get familiar with the notation of writing this the derivative of H of X with respect to X at x equals 9 this is equivalent this is equivalent to H let me do it in that blue color it is equivalent to H Prime and the prime signifies that we're doing the derivative H prime of H prime of X when x equals 9 so H prime of 9 is what this really is actually me do this in a different color so this is H prime of 9 so let's think about what that is let's take the derivative of both sides of this expression to figure out what the derivative with respect to X of H is so we get a derivative let me do that same white color derivative with respect to X with respect to X of H of X H of X is going to be equal to the derivative with respect to X of all of this business so I can actually just swell just rewrite it 3 times f of X plus 2 times G of X plus 2 times G of X now this right over here this the derivative of the sum of two terms that's going to be the same thing as the sum of the derivatives of each of the terms so this is going to be the same thing as the derivative with respect to X of three times right that a little bit neater three times f of X plus the derivative with respect to X of 2 times G of X two times G of X now the derivative of a number or I guess you could say a scaling factor times a function the derivative of a scalar times the function is the same thing as the scalar times the derivative of the function what does that mean well that just means that this first term right over here that's going to be equivalent to three times the derivative with respect to X of F of our f of X plus plus this part over here is the same thing as two we can make sure I don't run out of space here plus two times the derivative with respect to X the derivative with respect to X of G of X of G of X of G of X so the derivative of H with respect to X is equal to three times the derivative of F with respect to X plus two times the derivative of G with respect to X and if we wanted to write it in this kind of prime notation here we could rewrite it as H H prime of X is equal to three times F prime of X so this whole this part right over here that is the same thing as F prime of X so it's three times F prime of X plus two times G prime of X plus two times G prime of X and once you kind of are more fluent with this property the derivative of the sum of two things is the sum of the derivatives and the derivative of a scalar times something is the same thing as the scalar times the derivative of that something you really could have gone straight from straight from here to here pretty quickly now why is this interesting well now we can evaluate this function when X is equal to nine so H prime of nine is the same thing as three times F prime of nine plus two times G prime nine now what is F prime of nine the derivative of our function f when X is equal to nine well they tell us when X is equal to nine f of nine is one but more importantly f prime of nine is three so this part right over here evaluates that parts three but what's G prime of nine so let's look at this function a little bit a little bit more closely so there's a couple of ways we could think about it let's actually let's try to graph it now I think that could be interesting just to visualize what's going on here so let's say that's our y-axis and this right over here is our x-axis now when does an absolute value function like this what is this going to hit a minimum point well the absolute value of something is always going to be non-negative so it hits a minimum point when this thing when this thing is equal to zero well what is this thing equal to zero when x equals one this thing is equal to zero so we hit a minimum point when x is equal to one and when x equals one this term is zero absolute value of zero zero G of 1 is 1 so we have this point right over there now what happens after that what happens for X greater what happens for X greater than 1 actually let me write this down so G of X is equal to and in general whenever you have an absolute value of a relatively simple absolute value function like this you could think of it you could break it up into two function or you could think about this function over different intervals when the absolute value is non-negative and when the absolute value is negative so when the absolute value is non-negative that's when X is greater than or equal to zero X is greater than or equal to zero and when the absolute value is non-negative if you're taking the absolute value of a non-negative number then it's just going to be itself the absolute value of zero zero absolute value of 1 is 1 absolute value of 100 is 100 so then you could ignore the absolute value for X is greater than or equal to oh not greater than or equal to zero for X is greater than or equal to X is greater than or equal to one X is greater than or equal to one this thing right over here is non-negative and so it'll just evaluate to X minus 1 so this is going to be X minus 1 plus 1 which is the same thing as just X minus 1 plus 1 they just cancel out now when when X when this term right over here is negative and that's going to happen for X is this is going to happen for X is less than 1 well then the absolute value is going to be the opposite of it you give me the absolute value of a negative number it's going to be the opposite absolute value of negative 8 is positive 8 so it's going to be the negative of X minus 1 is 1 minus X 1 minus X plus 1 or you could say 2 minus X 2 minus X now so for X is greater than or equal to 1 we would look at this expression now what's the slope of that well the slope of that is 1 so we're going to have a curve that looks like or a line I guess we could say that looks like this that looks like this for all X is greater than or equal to 1 so the important thing remember we're going to think about we're going to think about the slope of the tangent line when we think about the derivative of G so slope is equal to 1 and for X less than 1 well our slope now if we look right over here our slope is negative our slope is negative 1 so it's going to look like this it's going to look it's going to look like that but for the point in question if we're thinking about G prime of 9 so 9 is some place out here so what is G prime of 9 so G prime of nine let me make it clear this graph right over here this is the graph of G of X or we could say Y this is the graph y equals G of X Y is equal to G of X so what is G prime of 9 well that's the slope when X is equal to 9 well the slope is going to be equal to 1 so G prime of 9 G prime of 9 is 1 so what does this evaluate to this is going to be 3 times 3 so this part right over here is 9 plus 2 times 1 +2 which is equal to 11 so the slope of the tangent line of H when X is equal to 9 is 11