Find $g'(f(2)),$ given $f(x)=\sqrt{x^2+5}$ and $g(x)=x^2+x$.
My first step was to find $g(f(x)),$ simplify if possible, then find the derivative $g'(f(x)).$ After I did this, I substituted $2$ for $x.$
A few other students we were saying to find the $g'(x),$ then find $f(2),$ then plug it into $g'(x).$ This also makes sense to me but I'm really unsure.
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Find $g'(f(2)),$ given $f(x)=\sqrt{x^2+5}$ and $g(x)=x^2+x$.
A few other students we were saying to find the $g'(x),$ then find $f(2),$ then plug it into $g'(x).$
My first step was to find $g(f(x)),$ then find the derivative $g'(f(x)).$ After I did this, I substituted $2$ for $x.$
The boldfaced part is somewhat incoherent:
If you indeed mean to find the derivative $g'(\color{red}{f(x)})$ and plug in $x=2,$ then you are obtaining the derivative $g'(\color{red}u)$ and plugging in $u=f(x)$ and plugging in $x=2,$ which this gives the same result as what your classmates are doing. That is, $$g'(\color{red}{f(2)}).$$
On the other hand, if you actually mean to find the derivative of $g(f(\color{red}x))$ and plug in $x=2,$ then you are obtaining $$(g\circ f)'(\color{red}2),$$ which gives a different result from the above.
The notation $g'(f(2))$ means the function $g'$ evaluated at the point $f(2)$. So the other students are right.