3 is just a constant, not one of the composited functions. We have:
and
The derivatives are:
and
The chain rule states that the derivative is . So the derivative is:
Hello, droidus!
Why is the derivative of not 0?
I thought you take the derivative of the outside (the derivative of 3 is 0), times the inside.
WOW! . . . I hope you are on your first day of differentiation.
Otherwise, you need some serious reviewing of the basics.
Would you say that the derivative of is zero
. . because the derivative of 3 is zero . . . times the "inside"?
Then any expression with a number in front has a derivative of zero?
You are trying (unsuccessfully) to use the Prouct Rule.
We have: .
Using the Product Rule: .
Obviously, using the Product Rule when one factor is a constant is a waste of energy
Did you ever use the Quotient Rule just to make it harder?
Example: .
Re-write it as: .
Now use the Quotient Rule: .
. . Whee! .Wasn't that fun?
when we multiply by a constant, we are employing the function:
f = c*(___), that is:
f(x) = cx.
this is a *different function* than f(x) = c, which has derivative 0.
let's find the derivative of f(x) = cx, straight from the definition:
so the derivative of the function f(x) = cx is the number c.
NOW we can use the chain rule.
suppose . then , where:
note that h itself is a composition: , where:
.
we'll get to h later, first let's use the chain rule on g:
as we saw above, , no matter what h(x) is (f' is a constant function).
so .
now we use the chain rule on h:
by the power rule, we have , so .
finally, we find p'(x) = 0 - 8 = -8.
now we "assemble the pieces":
.
Just in case a picture helps...
Ragnarok:
... where (key in spoiler) ...
Spoiler:
Halls:
... where (key in spoiler) ...
Spoiler:
Deveno:
More...
Spoiler:
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