Question

Given that the derivative $f^{\prime }\left(a\right)$ exists. Indicate which of the following statements (s) is/are always true-

• A. $f^{\prime }\left(a\right)={lim}_{h\to a}\frac{f\left(h\right)-f\left(a\right)}{h-a}$
• B. $f^{\prime }\left(a\right)={lim}_{h\to 0}\frac{f\left(a\right)-f(a-h)}{h}$
• C. $f^{\prime }\left(a\right)={lim}_{t\to 0}\frac{f(a+2t)-f\left(a\right)}{t}$
• D. $f^{\prime }\left(a\right)={lim}_{t\to 0}\frac{f(a+2t)-f(a+t)}{2t}$

Answer 1

Answer: (A), (B)

The correct options are
A $f^{\prime }\left(a\right)={lim}_{h\to a}\frac{f\left(h\right)-f\left(a\right)}{h-a}$
B $f^{\prime }\left(a\right)={lim}_{h\to 0}\frac{f\left(a\right)-f(a-h)}{h}$

The second option gives the definition of $f^{\prime }\left(x\right)$ and so it is correct.
Now, let us take $h=a+k$
Thus, $f^{\prime }\left(a\right)=li{m}_{k\to 0}\frac{f(a+k)-f\left(a\right)}{k}=li{m}_{h\to a}\frac{f\left(h\right)-f\left(a\right)}{h-a}.$
Hence option a is also correct.
However, the third option must look like $f^{\prime }\left(a\right)=li{m}_{t\to 0}\frac{f(a+2t)-f\left(a\right)}{2t}$ and the fourth must look like $f^{\prime }\left(a\right)=li{m}_{t\to 0}\frac{f(a+2t)-f(a+t)}{t}$ instead of the one given.