Question

Given the function f(x) = x^4 - 5x + 2 . Which of the following is the slope when x=2?
a. 22
b. 7
c. 28
d. -28

PLEASE HELP

Answer 1

Do you know how to take the derivative of f(x)?

Answer 2

You just need to take the derivative, then plug x=2 into the derivative. that gives you the slope of the graph at x=2.

Answer 3

@DebbieG hello when i plugged x into the equation i got an answer of 4 which isnt one of the choices so im messing up somewhere

Answer 4

What did you plug x=2 into? You said "the equation"... you aren't supposed to plug x=2 into f(x) (that just gives you f(2)). (I'm not sure if that's what you did though, because f(2) is not =4).

You need to first take the derivative of the f(x). Did you do that? What did you get?

Answer 5

\[f\prime(x) = \frac{ d }{ dx } (x^4 - 5x + 2)\]

Answer 6

Given that you are expected to know how to do this problem, I'm assuming that you've covered derivatives, at least for this type of function?

Answer 7

@DebbieG well this is all new to me we received homework after a class session where all this was introduced including derivatives

Answer 8

Hey, wait a second... the correct answer isn't there. Are you sure you've given the function correctly?

Answer 9

is the middle term -5x^2, by chance?

Answer 10

yes its x^4 -5x + 2 thats the whole equation

Answer 11

It's fine that it's new... you have to start somewhere. :) But it's important that you understand the rules for taking the derivative of a polynomial before you can tackle this problem.

Answer 12

Well then the answer is not there, but if instead, the function is

\(f(x)=x^4 - 5x^2 + 2\)

then the answer IS there. So I think that's a typo, and we should proceed with that assumption.

Answer 13

@DebbieG okay well can you help me through the steps of solving it please so i can know exactly how the answer was developed

Answer 14

Now to differentiate any term of a polynomial, e.g., anything that looks like \(5x^8\) or \(-2x^4\) or \(x^2\) --- anything where you have a constant and then a power of x - the rule is simple:

Pull the power out in front, multiply it by the constant (note that the constant might be 1), and then subtract one from the power.

E.g.:
\(\Large \frac{ d }{ dx } 3x^2=2\cdot3x^{2-1}=6x\)
\(\Large \frac{ d }{ dx } (-5x^4)=4\cdot-5x^{4-1}=-20x^3\)
\(\Large \frac{ d }{ dx } (x^6)=6\cdot x^{6-1}=6x^5\)

Now look those over, do you understand the rule? Do you see how I computed each derivative?

Answer 15

Also notice:
\(\Large \frac{ d }{ dx } (2x)=2\cdot x^{1-1}=2x^0=2\) (since \(b^0=1\))

So in other words, the derivative of a constant times x is just whatever the constant is.

And the last rule you'll need is this:

\(\Large \frac{ d }{ dx } c=0\) for any constant \(\large c\), e.g., the derivative of a constant is = 0.

Answer 16

Also, for polynomials, you get the derivative of the whole polynomial by just adding up the derivatives of each term... e.g.:

\(\Large f\prime(x) = \frac{ d }{ dx } (x^4 - 5x^2 + 2)= \frac{ d }{ dx } (x^4)+ \frac{ d }{ dx } (- 5x^2)+ \frac{ d }{ dx } (2)\)

Just take each derivative and then add them up. Remember, we are assuming that middle term should be an \(x^2\) term.

Answer 17

Now can you put that all together and tell me what our \(\Large f\prime(x)\) is? E.g., what do you get from:

\(\Large \frac{ d }{ dx } (x^4)+ \frac{ d }{ dx } (- 5x^2)+ \frac{ d }{ dx } (2)\)

Answer 18

@debbieG is f'(X)=7 ?

Answer 19

No. First of all, f'(x) here will be a FUNCTION OF x. Not a constant. (The only time it would be a constant is if you just have a linear expression, so the only x in the function is a 1st degree term, e.g., f(x)=7x+1. Then by the differentiation rules I gave you above, f'(x)=7.

Can you tell me, step by step, what you did? I want to see where you went wrong. Did you understand the rules I gave you above?

Answer 20

We want
\(\Large \frac{ d }{ dx } (x^4)+ \frac{ d }{ dx } (- 5x^2)+ \frac{ d }{ dx } (2)\)

So start with the first piece: what is \(\Large \frac{ d }{ dx } (x^4)\)? Just read up above, I told you exactly how to do it and showed you several examples.

Answer 21

It's most similiar to my last example, where I did:

\(\Large \frac{ d }{ dx } (x^6)=6\cdot x^{6-1}=6x^5\)

Answer 22

Now you do \(\Large \frac{ d }{ dx } (x^4)\)

Answer 23

@DebbieG is that the correct shown above

Answer 24

Right, so that's the derivative of the first term.

\(\Large \frac{ d }{ dx } (x^4)=4x^3\)

Now try the second term, same idea:
\(\Large \frac{ d }{ dx } (- 5x^2)=??\)