Find the derivative of the function f(x)=4x2−9xf(x)=4x^2-9xf(x)=4x2−9x.

f′(x)=8x−$$9f^{\prime}$$\left(x\right)=8x-9f′(x)=8x−9

Find the derivative of the function f ( x ) = 4 x 2 − 9 x f(x)=4x^2-9x f ( x ) = 4 x 2 − 9 x .

Let's see how we can solve this problem. So in this problem, we're asked to find the derivative of this function, f of x is equal to 4x squared minus 9x. Now the equation we need to find the derivative looks like this. F prime of x is going to be equal to the limit as h approaches 0, and then the equation is going to be f of x plus h minus f of x all divided by h. And this is the equation that we've learned about. So let's go ahead and see if we can use this equation here and apply it to the function that we have. Well, this limit as h approaches 0, this is going to stay consistent. We're going to keep this, and then what we're gonna have is f of x plus h. And what I can see here is that's going to be this function with x+h plugged in for x. So we're going to have 4 times x+hsquared, and that's going to be minus 9 times x+h. And then here, we're going to subtract off f of x. Well, f of x is just going to be this, The rest of this function we have is just gonna be this function repeated, so we'll have 4x squared minus 9x. And I'll go ahead and put this in parentheses so we can separate these 2. And all of that's gonna be divided by h. So this is what things look like. Now at this point, again, I can't really apply this limit because this h here would be 0 and we can't divide by 0, so I'm gonna need to factor and somehow get this h out of the denominator. Now we're gonna keep this limit attached, and at this point, I'm going to try to simplify what we have above. Now what I can see here is we have this 4 times x plus h squared. X plus h squared is going to be the same thing as x squared plus 2xh plus h squared. That's what happens when you foil everything out. So that's just x plus h times x plus h. Then we're going to have minus, and I'm going to go ahead and distribute this 9 into the parenthesis here. So we're going to have minus 9 x, and then it's a negative 9 here. So we'll have minus 9 h as well. And then all of that is going to be and then I'll go ahead and distribute this negative sign in here. So So going to have minus 4x squared, and then a negative times a negative would give you plus 9x. This is what we have. What I will do also is distribute this 4 into the parenthesis here. So going up here, and I'll continue this f prime of x here is going to be equal to the limit as h approaches 0. And then distributing this forward, we're gonna have 4x squared plus and then 4 times 2, which is 8xh plus 4h squared. And we're gonna have minus 9x minus 9h. We have minus 4x squared plus 9x. And all of that is going to be divided by h. Now let's see what we can cancel here. Well, I can see that we have a positive 4x squared and a negative 4x squared, as you have a negative 9x and a positive 9x. And then let's see. And I don't know if there's anything we have with this h or any of these, so I believe we've canceled all the terms that we can. So what that means is that we're going to have the limit as h approaches 0. This limit stays on, and that's going to be 8xh plus 4h squared minus 9h all divided by h. Now what I can do from here is I can actually factor out this h that I see in each of these terms. So, again, we're going to have this limit continuing. So we'll have limit as h approaches 0, and this is going to be 8 time or excuse me. H times 8x+4hminus9 all divided by h, in which case this h is going to cancel. So now we're in business. All we need to do is take this 0 here and plug it in for h. So we're going to have that this function f prime of x is equal to 8x+4 times and then we're replacing this h with 0, so 4 times 0 minus 9, which is all just equal to 8x minus 9. So this right here would be the solution and the derivative, and that is how you can solve this problem. So I hope you found this video helpful, and let's move on.