We use a variety of different notations to express the derivative of a function. In Example 3.12 we showed that if 𝑓(𝑥)=𝑥2−2𝑥, then 𝑓′(𝑥)=2𝑥−2. If we had expressed this function in the form 𝑦=𝑥2−2𝑥, we could have expressed the derivative as 𝑦′=2𝑥−2 or 𝑑𝑦𝑑𝑥=2𝑥−2. We could have conveyed the same information by writing 𝑑𝑑𝑥(𝑥2−2𝑥)=2𝑥−2. Thus, for the function 𝑦=𝑓(𝑥), each of the following notations represents the derivative of 𝑓(𝑥): 𝑓′(𝑥),𝑑𝑦𝑑𝑥,𝑦′,𝑑𝑑𝑥(𝑓(𝑥)). In place of 𝑓′(𝑎) we may also use 𝑑𝑦𝑑𝑥∣∣∣𝑥=𝑎 Use of the 𝑑𝑦𝑑𝑥 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 Δ𝑦Δ𝑥 where Δ𝑦 is the difference in the 𝑦 values corresponding to the difference in the 𝑥 values, which are expressed as Δ𝑥 (Figure 3.11). Thus the derivative, which can be thought of as the instantaneous rate of change of 𝑦 with respect to 𝑥, is expressed as 𝑑𝑦𝑑𝑥=limΔ𝑥→0Δ𝑦Δ𝑥.