Question

help me derivatives again, f(x)= (2x^2 + 5)^3 (4x-1) diff calculus. can you show me the proper way to solve to easy to understand. please

Answer 1

\[f(x) =(2x^2+5)^3(4x-1)\]You could... use the product rule to find the derivative, perhaps. \[d(f(x)) = f'g +g'f\]

Answer 2

so how can we solve that?

Answer 3

\[d(f(x)) = [(2x^2+5)^3]'\cdot (4x-1) +(4x-1)'\cdot (2x^2+5)^3\]

Answer 4

Can you take it from there?

Answer 5

ill get my reading glass

Answer 6

Hows it going? :)

Answer 7

how can we solve that?

Answer 8

Do you know how to use the chain rule?

Answer 9

no

Answer 10

\[d(x^n) = nx^{n-1}\]This?

Answer 11

no sorry. :((

Answer 12

Chain rule works like this: \[d(f(g(x)) = [f(x)]' \cdot [g(x)]'\]So.. you have a function within a function. \[f(x) = (2x^2+5)^3\]\[g(x) = 2x^2+5\]Are you with me so far?

Answer 13

yes im with you :)

Answer 14

so what our the final answr?

Answer 15

Well, I am guiding you through to find it :)

Answer 16

but its hard

Answer 17

Yes, calculus is quite difficult.

Answer 18

Stay with me here though, you will learn how to do this :)

Answer 19

yes mam/ :))

Answer 20

So first we take the derivative of our "outer" function, \(f(x)\). In order to do that, we apply the power rule, \(\large \frac{d}{dx}(x^n) = n\cdot x^{n-1}\). Basically, we bring down the power of the function, and reduce it by 1. So \[\frac{d}{dx} (f(x)) = 3(2x^2+5)^{3-1} = 3(2x^2+5)^2\]

Answer 21

Did you understand that?

Answer 22

ah i try to get it

Answer 23

so our answr should be 3(2x^2+5)^2

Answer 24

That is not our answer yet, that is just a part of it.

Answer 25

ahh sorrry mam

Answer 26

Now we take our "inner" function, and we also the power rule to it. Then we multiply it to the derivative of our function, \(f(x)\)\[\frac{d}{dx}(g(x)) = (2x^2+5)' = 4x\]

Answer 27

So for the first portion, we will have \[\frac{d}{dx}(2x^2+5)^3) = 3(2x^2+5)^2\cdot4x = 12x(2x^2+5)^2\]

Answer 28

Are you undertanding any of this?

Answer 29

yes mam

Answer 30

so if we get the power role? mam what will happen to our solving?

Answer 31

I don't understand your question?

Answer 32

what will happen to our solving? if we get the power rule

Answer 33

The power rule is the method we use to find derivatives, it helps make functions easier to combine and solve for.

Answer 34

But anyway,
So we started with this: \[d(f(x)) = \color{red}{[(2x^2+5)^3]'}\cdot (4x-1) +(4x-1)'\cdot (2x^2+5)^3\]And that highlighted portion we found, turned into this:So we started with this: \[d(f(x)) = [\color{red}{12x(2x^2+5)^2}\cdot (4x-1) +(4x-1)'\cdot (2x^2+5)^3\]

Answer 35

yes mam

Answer 36

Now we do the same thing with our \(g(x)\) function, \((4x-1)\).\[\frac{d}{dx} (4x-1) = \frac{d}{dx}(4x) = 4\]

Answer 37

yes mam

Answer 38

\[d(f(x)) = [12x(2x^2+5)^2\cdot (4x-1) +\color{red}{(4x-1)'}\cdot (2x^2+5)^3\]\[d(f(x)) = 12x(2x^2+5)^2\cdot (4x-1) +\color{red}4\cdot (2x^2+5)^3\]

Answer 39

So now, we see that \((2x^2+5)^2\) is common throughout the function, we can factor that out.\[d(f(x)) = 12x(2x^2+5)^2\cdot (4x-1) +4\cdot (2x^2+5)^3\]\[f'(x)=(2x^2+5)^2[12x(4x-1)+4(2x^2+5)]\]

Answer 40

Now you can simplify what is in the brackets and see if anything cancels out, or combines, or you can leave it that way.

Answer 41

yes mam

Answer 42

before doing any more derivatives, read through your calculus book to learn how to find them. That will come in handy.

Answer 43

Good luck :)

Answer 44

mam

Answer 45

last question!

Answer 46

yes?

Answer 47

how can i cet the final answr?

Answer 48

Simplify what is inside the brackets.

Answer 49

how>?

Answer 50

I have worked most of the problem for you, I am sure you know how to combine like terms :)

Answer 51

mam

Answer 52

I have got to go now, I hope you practice these more :)

Answer 53

thank you, :)) You know I was thinking, I don't want you to be my math teacher. I don't want you to teach me. Because I like you so much.

Answer 54

Good luck :)

Answer 55

just good luck? @Jhannybean