Question

1. if $f(2) = 4$, then find $limlimits_{delta x \to 0} \frac{f(2 - 2delta x) - f(2)}{delta x}$.

Explanation:

Step1: Recall the definition of derivative

The derivative of a function \( f(x) \) at a point \( a \) is defined as \( f^{\prime}(a)=\lim_{\Delta x
ightarrow0}\frac{f(a + \Delta x)-f(a)}{\Delta x} \). In our problem, we have \( \lim_{\Delta x
ightarrow0}\frac{f(2-2\Delta x)-f(2)}{\Delta x} \), let's make a substitution. Let \( h=- 2\Delta x \), then when \( \Delta x
ightarrow0 \), \( h
ightarrow0 \), and \( \Delta x=-\frac{h}{2} \).

Step2: Substitute the variable

Substitute \( h = - 2\Delta x \) (or \( \Delta x=-\frac{h}{2} \)) into the limit:

$$
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$$

Step3: Use the derivative definition

We know that \( f^{\prime}(2)=\lim_{h
ightarrow0}\frac{f(2 + h)-f(2)}{h} = 4\). Substitute this into the above expression:
\(-2\times f^{\prime}(2)=-2\times4=-8\)

Answer:

\(-8\)