Answer:
We use a variety of different notations to express the derivative of a function. In (Figure) we showed that if f(x)=x^2-2x, then f^{\prime}(x)=2x-2. If we had expressed this function in the form y=x^2-2x, we could have expressed the derivative as y^{\prime}=2x-2 or \frac{dy}{dx}=2x-2. We could have conveyed the same information by writing \frac{d}{dx}(x^2-2x)=2x-2. Thus, for the function y=f(x), each of the following notations represents the derivative of f(x):
f^{\prime}(x), \, \frac{dy}{dx}, \, y^{\prime}, \, \frac{d}{dx}(f(x)).
In place of f^{\prime}(a) we may also use \frac{dy}{dx}\Big|_{x=a} Use of the \frac{dy}{dx} notation (called Leibniz notation) is quite common in engineering and physics. To understand this notation better, recall that the derivative of a function at a point is the limit of the slopes of secant lines as the secant lines approach the tangent line. The slopes of these secant lines are often expressed in the form \frac{\Delta y}{\Delta x} where \Delta y is the difference in the y values corresponding to the difference in the x values, which are expressed as \Delta x ((Figure)). Thus the derivative, which can be thought of as the instantaneous rate of change of y with respect to x, is expressed as
\frac{dy}{dx}=\underset{\Delta x\to 0}{\lim}\frac{\Delta y}{\Delta x}.
Explanation: