The problem asks to identify the correct definition of the derivative from first principles. The options are variations of the limit definition of the derivative.

1. Problem Description

The problem asks to identify the correct definition of the derivative from first principles. The options are variations of the limit definition of the derivative.

2. Solution Steps

The derivative of a function $f(x)$ from first principles (also known as the limit definition of the derivative) is defined as:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Here, $h$ is a small change in $x$, which is often represented as $\delta x$ in some notations. Replacing $h$ by $\delta x$, we have:

$f'(x) = \lim_{\delta x \to 0} \frac{f(x+\delta x) - f(x)}{\delta x}$

In the given question, the options are expressed in terms of $\frac{dy}{dx}$. The derivative is the same as $f'(x)$, so we have:

$\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{f(x+\delta x) - f(x)}{\delta x}$

Now, we compare the given options to this formula.

(A) $\frac{dy}{dx} = \lim_{\delta x \to \infty} \frac{f(x+\delta x) + f(x)}{\delta x}$ is incorrect. The limit should approach 0, not infinity, and it should be a difference, not a sum, in the numerator.

(B) $\frac{dy}{dx} = \lim_{\delta x \to \infty} \frac{f(\delta x) - f(x)}{\delta x}$ is incorrect. The function evaluated at $x+\delta x$ is needed, and the limit should approach

0. (C) The option is cut off and we cannot see it fully.

(D) $\frac{dy}{dx} = \lim_{\delta x \to \infty} \frac{f(x+\delta x) - f(x)}{f(\delta x)+ f(x)}$ is incorrect. The denominator should be $\delta x$ and the limit should approach

0.

None of the given options are correct with the limit tending to infinity. However, we need to choose the 'best' answer, by assuming there is a typographical error. The correct formula has $\delta x \to 0$ and $\frac{f(x+\delta x) - f(x)}{\delta x}$.

Option (B) has most of it right. So let us assume the limit should be $\delta x \to 0$. Also assume the numerator in (B) should read $f(x+\delta x) - f(x)$. Then it would be correct.

Therefore, based on the options, (B) is the closest to the correct definition of the derivative from first principles if we assume that the limit approaches 0 and it contains $f(x+\delta x)$ rather than $f(\delta x)$.

3. Final Answer

(B)