Finding out the derivatives through table values.

Question

Question: Suppose that the functions $f$ and $g$ and their derivatives with respect to x have the following values at $x = 1$ and $x = 2$.

My Answer: Part 1 Answer Part 2 Answer

My Problem: My friend said that you have to take out the half from $x/2$ when you derive it. But I don't think that's the proper way to do the chain rule. Who do you think is right over this matter?

Answer 1

When taking derivatives at a point, say $x=2$, it is very important that you only plug in $2$ after taking the derivative.

See for example the first part

$$\begin{align} \frac{d}{dx}(f(x/2)g(x)) &=\frac{d}{dx}\Big(f(x/2)\Big)g(x)+f(x/2)\frac{d}{dx}\Big(g(x)\Big)\\ &=f'(x/2)\color{blue}{\frac{d}{dx}\Big(x/2\Big)}g(x)+f(x/2)g'(x)\\ &=\color{blue}{\frac{1}{2}}f'(x/2)g(x)+f(x/2)g'(x) \end{align}$$

Now we plug in $x=2$ to get the correct answer for part (i). Does this make sense?