Umm, i still dont fully understand how you get that derivitive, i realize now its not the product rule as it is a function inside a function so it must be the chain rule?
Can you run me through what you did?
That is the chain rule. Think about what you are doing with the chain rule though, to differentiate
^3)
, you are essentially doing this
^3\cdot \frac{d}{dx}(x+3)=3(x+3)^2\cdot (1)=3(x+3)^2)
. Notice the "chain" of finding the derivative of the outside function, then the inside function, works the same with 3 functions inside eachother
e.g.
^2+1)^2)
, same principle applies, take the derivative of the outside function, then the next function, then the next
^2+1)^2\cdot \frac{d}{dx}\left [ (x+3)^2+1 \right ]\cdot \frac{d}{dx}(x+3)=2((x+3)^2+1)\cdot \left [ 2(x+3)\right ] \cdot (1)=\left [ 2(x+3)^2+2\right ](2x+6)=4 x^3+36 x^2+112 x+120)
Now using that with the question.
\right ])
is a function with a function, so we need to differentiate like we did before, in order of nested functions.
\right ]\cdot \frac{d}{dx}g(x))
, which turns out to be
\right ]\cdot g'(x))
Not sure how that helped, just run through more of those nested derivatives (functions in functions in functions etc.) and you'll see what I mean. Also think about it in terms of Leibniz notation (I seem to understand the chain rule better with his notation)
and now let's assume it was with the function before (the 3 functions within eachother) it becomes
where
^2+1)
and
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