chain rule

[tlc]

we have this table showing values:

then some equations we have to find the derivitive of.
i cannot for the life of me figure out what to do for these two:


1. f (x * f(x)) when x = -4

&

2. f(x) when x= -16
g(x) + 6

can someone help me?

i have tried:
1. f'(x * f(x))* x * f'(x) which didnt work

 

[soroban]

Hello, tlc!

$\displaystyle \;\;\;x\;\;\,f(x)\;\,g(x)\;\,f'(x)\;\,g'(x)$

$\displaystyle \;-16\;\;\;16\;\;\;64\;\;\;7\;\;\;\;\,3$

$\displaystyle \;\,-4\;\,-16\;\;\;16\;\;\;\:2\;\;\;-4$

$\displaystyle \;\;\;0\;\;-2\;\;\;\;4\;\;\;\;1\;\;\;\;7$

$\displaystyle \;\;16\;\;\;\:3\;\;\;\;6\;\;\;-2\;\;\;-1$

$\displaystyle \;\;64\;\;\;0\;\;\;\;\,2\;\;\;-2\;\;\;-2$

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1. Given: $\displaystyle \;y\;=\;f(x\cdot f(x))$. . . Find $\displaystyle y'$ when $\displaystyle x\,=\,$-$\displaystyle 4$

Use the Chain Rule and the Product Rule, then substitute the values.

$\displaystyle y'\;=\;f'[x\cdot f(x)]\cdot\left[x\cdot f'(x)\,+\,f(x)\right]$

$\displaystyle y'\;=\;f'[$-$\displaystyle 4\cdot f($-$\displaystyle 4)] \cdot [$-$\displaystyle 4\cdot f'($-$\displaystyle 4)\,+\,f($-$\displaystyle 4) ]$

$\displaystyle y'\;=\;f'[ ($-$\displaystyle 4)($-$\displaystyle 16)] \cdot [$-$\displaystyle 4(2)\,+\,($-$\displaystyle 16)]$

$\displaystyle y'\;=\;f'(64)\cdot[$-$\displaystyle 8\,-\,16]$

$\displaystyle y'\;=\;($-$\displaystyle 1)\cdot[$-$\displaystyle 24]$

$\displaystyle y'\;=\;24$

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2. Given: $\displaystyle \;y \;= \;\frac{f(x)}{g(x)\,+\,6}$ . . Find $\displaystyle y'$ when $\displaystyle x\,=\,$-$\displaystyle 16$

Use the Quotient Rule, then substitute the values.

\(\displaystyle \L y'\;=\;\frac{\left[g(x)\,+\,6\right]\cdot f'(x)\,-\,f(x)\cdot g'(x)}{\left[g(x)\,+\,6\right]^2}\)

\(\displaystyle \L y'\;=\;\frac{[g(-16)\,+\,6]\cdot f'(-16)\,-\,f(-16)\cdot g'(-16)}{[g(-16)\,+\,6]^2}\)

\(\displaystyle \L y'\;=\;\frac{[64\,+\,6]\cdot(7)\,-\,(16)(3)}{(64\,+\,6)^2}\)

\(\displaystyle \L y'\;=\;\frac{(70)(7)\,-\,(16)(3)}{(70^2)}\)

\(\displaystyle \L y'\;=\;\frac{490\,-\,48}{4900}\;=\;\frac{442}{4900}\;=\;\frac{221}{2450}\)