## haleyelizabeth2017 one year ago Derivatives of quadratic equations :)

1. freckles

$f(x)=ax^2+bx+c \\ \text{ so you want to differentiate } f$ First question: Are you doing short cuts or the definition way?

2. haleyelizabeth2017

I'm not sure, I just would like to learn how to do this so I'm not always confused lol

3. freckles

You are probably doing definition I bet.

4. freckles

$f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \text{ or } f'(x)=\lim_{z \rightarrow x} \frac{f(z)-f(x)}{z-x}$ you can use either one of these these are basically the same thing they are the same thing it can be shown with a substitution Let's play with the first one...

5. freckles

$f(x)=ax^2+bx+c \\ f(x+h)=a(x+h)^2+b(x+h)+c$ To find f(x+h) notice I just replace all the x's in the first expression with (x+h)'s.

6. freckles

the numerator in my definition is f(x+h)-f(x) so I could go ahead and look at simplifying that first I will state what is without the simplification : $f(x+h)-f(x)=[a(x+h)^2+b(x+h)+c]-[ax^2+bx+c]$ The next part is to expand and cancel.

7. freckles

Can you expand (x+h)^2

8. haleyelizabeth2017

$x^2+2hx+h^2$

9. freckles

cool so I'm going to put that in place of (x+h)^2 ... $f(x+h)-f(x)=[a(x^2+2xh+h^2)+b(x+h)+c]-[ax^2+bx+c] \\ \text{ now distributing a bit } \\ f(x+h)-f(x)=ax^2+2axh+ah^2+bx+bh+c-ax^2-bx-c$ do you have a problem with my distributing step?

10. haleyelizabeth2017

Nope, easy peasy

11. freckles

you should see some things that cancel (or zero out) like ax^2-ax^2=0 and bx-bx=0 and c-c=0

12. freckles

$f(x+h)-f(x)=\cancel{ax^2}+2axh+ah^2+\cancel{bx}+bh+\cancel{c}-\cancel{ax^2}-\cancel{bx}-\cancel{c}$

13. haleyelizabeth2017

So we left with $f(x+h)-f(x)=2axh+ah^2+bh$

14. freckles

right!

15. freckles

now let's go back to our definition this was only the numerator

16. haleyelizabeth2017

Okay :)

17. freckles

$f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ f'(x)=\lim_{h \rightarrow 0} \frac{2axh+ah^2+bh}{h}$

18. freckles

so... notice that every term on top has a common factor of h and the denominator also has a common factor h

19. freckles

$f'(x)=\lim_{h \rightarrow 0} \frac{h(2ax+ah+b)}{h(1)}$

20. haleyelizabeth2017

Oh!

21. freckles

we can write as: $f'(x)=\lim_{h \rightarrow 0} \frac{h}{h} \cdot \frac{2ax+ah+b}{1}$

22. haleyelizabeth2017

which the h's cancel, right?

23. freckles

yep and that whole h on the bottom was the thing that was giving us problems but guess what?

24. freckles

it is no longer there because we canceled that mean h guy on bottom

25. haleyelizabeth2017

:)

26. freckles

so we can plug in 0 now

27. freckles

for h that is

28. haleyelizabeth2017

and it leaves us with 2ax+b?

29. freckles

$f'(x) =\lim_{h \rightarrow 0} (2ax+ah+b)=2ax+a(0)+b=2ax+0+b=2ax+b$ you are right

30. haleyelizabeth2017

Is that it?

31. freckles

$f(x)=ax^2+bx+c \text{ gives us } f'(x)=2ax+b$

32. freckles

yes now you can find the derivative of any quadratic

33. freckles

you actually have a formula for it now we just found a formula for it above

34. haleyelizabeth2017

Awesome! Didn't know it was that simple...the video I watched on it made it seem so confusing lol

35. haleyelizabeth2017

So there is never a c...

36. freckles

$f(x)=ax^2+bx+c \text{ gives } f'(x)=2ax+b \\ \text{ example } \\ \text{ say the question is find } f' \text{ given } f(x)=5x^2+3x-4 \\ \text{ we only need to know } a \text{ and } b \\ f'(x)=2(5)x+3=10x+3$

37. freckles

Well the derivative of a constant will always be zero.

38. haleyelizabeth2017

Schnazzy!

39. freckles

Because the slope of y=a constant is zero.

40. haleyelizabeth2017

Thank you so very much! :)

41. freckles

Np. So this is exactly what you wanted right? Do you have anymore questions about differentiating a quadratic?

42. haleyelizabeth2017

Yes and no, I don't! :)

43. haleyelizabeth2017

Wait...perhaps one question...this works for all quadratic equations?

44. freckles

Alright. I might do one more thing before I leave... A different strategy... Instead of expanding...grouping and factoring... $f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ f'(x)=\lim_{h \rightarrow 0} \frac{ a(x+h)^2+b(x+h)+c-(ax^2+bx+c)}{h} \\ f'(x)=\lim_{h \rightarrow 0} \frac{a[(x+h)^2-x^2]+b((x+h)-x]+[c-c]}{h} \\ f'(x)=\lim_{h \rightarrow 0} \frac{a[(x+h)^2-x^2]+b[(x+h)-x]+0}{h} \\ f'(x)=\lim_{h \rightarrow 0} \frac{a[(x+h-x)(x+h+x)+b[x+h-x]}{h} \\ \\ f'(x)=\lim_{h \rightarrow 0} \frac{ah(2x+h)+bh }{h} \\ f'(x)=\lim_{h \rightarrow 0} a(2x+h)+b \\ f'(x)=a(2x+0)+b =2ax+b$

45. freckles

and yes to your question you do know a quadratic function is in this form right: $f(x)=ax^2+bx+c ?$

46. haleyelizabeth2017

yes!

47. freckles

and if you do then yes to your question

48. haleyelizabeth2017

I love quadratic equations....and finding the solutions using the quadratic formula too :)

49. freckles

do you know that you can find the x-coordinate of vertex of the parabola using the derivative

50. haleyelizabeth2017

I did not!

51. freckles

Yep. You will probably learn this later. But You can use the derivative to either find the max or min of a function. You can set the derivative equal to zero and find where it does not exist...These will be critical numbers... So you have $f(x)=ax^2+bx+c \\ f'(x)=2ax+b \\ f'(x)=0 \text{ gives you the equation } 2ax+b=0 \\ \\ \text{ solving for } x \text{ gives you } x=\frac{-b}{2a}$

52. freckles

This is exactly what we would get if we did the whole completing the square thing.

53. freckles

$f(x)=ax^2+bx+c \\ f(x)=ax^2+\frac{a}{a}bx+c \\ f(x)=a(x^2+\frac{1}{a}bx)+c \\ f(x)=a(x^2+\frac{b}{a}x)+c \\ f(x)=a(x^2+\frac{b}{a}c+(\frac{b}{2a})^2)+c-a(\frac{b}{2a})^2 \\ f(x)=(x+\frac{b}{2a})^2+c-a(\frac{b}{2a})^2 \\ \text{ where the vertex is } (-\frac{b}{2a}, c-a(\frac{b}{2a})^2)$

54. freckles

But anyways all of that is extra.

55. haleyelizabeth2017

Thank you again :)

56. freckles

np