Question

if r(x)=f(-2x), r'(x)=?

Answer 1

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What does r'(x) actually mean?
It means what is the derivative of r(x) with respect to 'x' [Because 'x' is what is inside the brackets. Or, we are differentiating with respect to that.\[r'(x) = \frac{d[r(x)]}{dx}\]
Now input the value of r(x) = f(-2x) in the derivative.\[r'(x) = \frac{d[f(-2x)]}{dx}\]
Now use chain rule.
\[r'(x) = \frac{d[f(-2x)] ~.~ d(-2x)}{d(-2x) ~.~d(x)} = -2~.~f'(-2x)\]

Notice, here, \(f'(-2x)\) represents the derivative of 'f' w.r.t. (-2x) and not (x) here.

Getting this? :)