Question

Find the derivative of f(x) = 3x + 8 at x = 4.

Answer 1

Interestingly (though not really)... the derivative of a linear function is just the slope.

Answer 2

It's delta y/delta x as delta y and delta x get close to zero - but for a linear equation, the slope is always constant - aka the rate of change

Answer 3

these are the answers idk how to get it..
1

3

4

8

Answer 4

word of unrelated advice - learn calculus. Seriously, if you're in AB or BC and you decide for whatever reason not to pay attention during limits and derivatives, you're going to fail integrals completely

Answer 5

<le gasp>

First things first... the derivative of a sum (or difference) is the *sum (difference) of their derivatives*

Aye?

Answer 6

ya seriously lol i'm in pre calc..... and i think so

Answer 7

Okay, what's the derivative of a power?
\[\Large \frac{d}{dx}ax^n=\color{red}?\]

Answer 8

0?

Answer 9

Don't guess.
You seriously aren't familiar with the power rule? D:

Answer 10

nope i'm taking it online they don't teach anything

Answer 11

Here it is...
\[\Large \frac{d}{dx}ax^n\]
You take the exponent (n), multiply it to the function:
\[\Large \frac{d}{dx}ax^n\rightarrow \color{red}nax^n\]
BUT you subtract 1 from the exponent
\[\Large \frac{d}{dx}ax^n\rightarrow \color{red}nax^n\rightarrow nax^{\color{red}{n-1}}\]

Answer 12

And THAT... is the power rule:
\[\Large \frac{d}{dx}ax^n = \color{blue}{nax^{n-1}}\]

Answer 13

So, having said that, what is the derivative of
\[\Large \frac{d}{dx}3x = \frac{d}{dx}3x^1=\color{red}?\]

Answer 14

is it 1

Answer 15

I don't know, IS IT?
Check the formula I gave you.

Answer 16

im so confused

Answer 17

@luv2chatxoxo , have you gone over the "difference quotient" yet?

Answer 18

I don't know, IS IT?
Check the formula I gave you.

Answer 19

Okay, here's an example:
Suppose I want the derivative of \(\large 5x^4\).
So first, the exponent is 4, so I multiply that to the function, but I subtract 1 from the exponent... like so:
\[\Large \frac{d}{dx}5x^4 = \color{red}4\cdot5x^{4\color{red}{-1}}=\color{blue}{20x^3}\]

Answer 20

so is it 3x^0?

Answer 21

Precisely ^_^
And what's x^0?

Answer 22

Hey, you still there?
You're doing great...
now what is
\[\Large x^0=\color{red}? \]

Answer 23

1 sorry!

Answer 24

Good.
So the derivative of 3x is just 3.

What about the derivative of 8?

Clue:
\[\Large 8 = 8x^0\]

Apply the power rule again...

Answer 25

8

Answer 26

wait hold on

Answer 27

would it be 8x^-1?

Answer 28

You forgot something...
\[\Large \frac{d}{dx}ax^n = \color{red}nax^{n-1}\]

Answer 29

so would it just be 0?

Answer 30

That's precisely it :P

Keep that in mind.
The derivative of a *constant* is *always* zero.

So the derivative of 3x + 8
is just the sum of the derivatives of 3x and 8
Which are 3 and 0, respectively.

so finally, what's the derivative of 3x + 8?

Answer 31

3!!

Answer 32

right? haha

Answer 33

Yes.
And 3 is constant, so no matter what x is, be it 4, or literally anything on the domain, the derivative is always 3.

Answer 34

so we don't even have to do anything with the 4?

Answer 35

Nope.
You got lucky, the derivative of this one was a constant.

Just so I know you understand, what's the derivative of
\[\Large 4x^2+x\]at x = 4?

Answer 36

see now would it be 8x^1 + x^-1? and do we plug in four for x?

Answer 37

uh-oh...
how did you get x^-1?
The original exponent of x was 1...

Answer 38

crap i mean x^0

Answer 39

Which is...?

Answer 40

1

Answer 41

so would the answer be 36

Answer 42

uhh...
8x + 1?

Answer 43

Maybe you need to check that again...

Answer 44

oh wow 33

Answer 45

tsk tsk :P

One last, just to reassure me (and yourself) that you're well on your way to mastering the power rule.

Answer 46

i have another problem i can send you and i can see if i get it

Answer 47

this one!! Find the derivative of f(x) = 8x2 + 11x at x = 7.

Answer 48

Well, shoot.
Go ahead and do it.
Tell me your answer.

Answer 49

123?

Answer 50

Brilliant ^_^

Answer 51

Try this one too.
I have confidence in you...

Derivative of \[\Large x^3 +2x^2-6x+4\] at x = 1

Go for it, tiger :P

Answer 52

ah yay thank you!! quick question though

Answer 53

how would you solve this one? Find the derivative of f(x) = 5/x at x = -1.

Answer 54

do you just plug in -1 for x?

Answer 55

so it would be -1^0?

Answer 56

Hang on.
Step-by-step.
What's the derivative of 5/x ?
do you know?

Answer 57

noo

Answer 58

Pity :P
Let's write it this way:
\[\Large \frac5x = 5x^{-1}\]
does that look better? :P
Power rule still applies, by the way..

Answer 59

how do you get to the negative 1 though

Answer 60

like x raised to it

Answer 61

Seriously? LOL
laws of exponents...
\[\Large \frac1{x^n}= x^{-n}\]

Answer 62

haah ok ok i remember now ive had a long morning okay so then i plug in the -1 for x correct?

Answer 63

Not yet.
First find the derivative.

Answer 64

why am i getting so confsued?

Answer 65

Because... you didn't pay attention to basics? :P

Answer 66

nooo i just have to teach everything to myself lol so 5x^-1 idk what to do

Answer 67

I said... THIS
\[\Large \frac{d}{dx}ax^n = \color{blue}{nax^{n-1}}\]
still applies!

Even if the exponent is negative!

Answer 68

ohh!!!! so 5x^-2

Answer 69

No... remember that your n is negative!

Pay attention to signs!

Answer 70

so wouldnt that be -1-1? isn't that negative 2?

Answer 71

You can't keep forgetting these things!
\[\Large \frac{d}{dx}ax^n = \color{red}nax^{n-1}\]

Answer 72

-5x^-2?

Answer 73

Better.
Now plug in x = -1

Answer 74

so -5?

Answer 75

mhmm ^_^

Answer 76

one more :( i'm sorry i jsut need to make sure i get these Find the derivative of f(x) = -7/x at x = -3.