Why is $\delta x$ tending to be zero in dy/dx and not $\delta y$ tending to be 0?
Why is $\Delta x $ and not $\Delta y $ written to be tending towards 0 in case when we give limits to form dy/dx.
, $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \frac{dy}{dx}$$
Can we solve this equation like we solve when there is a = sign and we take denominator towards the other numerator? Like $$\frac{3}{4}=\frac{6}{8}$$
So we get 24=24.
, $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \frac{dy}{dx}$$
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$\begingroup$The derivative of a function is often described as the instantaneous rate of change. I will explain this by way of analogy. Say it takes you $10$ seconds to drive $100$m in a car. Your average speed is $10$m/s, but it is possible that during those $10$ seconds, sometimes you were going faster, and other times you were going slower. Even in the space of $0.1$ seconds, perhaps your speed changed a little bit during that time. So rather than trying to calculate one's speed over a specific time interval, we instead try to make rigorous the notion of what your speed is at one particular point. We do this by calculating the limit$$ \frac{ds}{dt}=\lim_{\Delta t \to 0}\frac{\Delta s}{\Delta t} \, . $$As $t$ gets closer and closer to $0$, there is less and less room for your speed to be changing during that time period. This can be visualised by zooming in on the velocity-time graph. It looks almost linear, meaning that, for example, in the space of $0.000001$ seconds, your speed—the rate of change of distance—is pretty much constant.
Because we are considering a limit, it is not possible to 'solve' the above equation in the way you would a fraction. This is because we are not considering any one value of $t$, but rather what happens at $t$ approaches $0$.
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