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# Calculus

## Deriving the Formula

Recall the formula for finding the slope of a line joining two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Suppose these two points are points on a curve $$y = f(x)$$:

Notice that $$y_1 = f(x_1)$$ and $$y_2 = f(x_2)$$. Let's adjust our notation slightly. Instead of $$x_1$$ and $$x_2$$, let's use $$x$$ and $$x+h$$ instead. Therefore, instead of $$y_1$$ and $$y_2$$, we now have $$f(x)$$ and $$f(x+h)$$ as shown below

Notice that with the new notation, the formula for the slope of the line has now become

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{f(x+h) - f(x)}{x+h-x} = \frac{f(x+h) - f(x)}{h}$

Now suppose we make these two points on the curve closer together (we do this by letting $$h$$ become very small).

The closer we push the second point towards the $$(x,f(x))$$, the closer the above formula comes to giving us the slope of the tangent line to the curve at the point $$(x,f(x))$$.

The expression

$\frac{f(x+h) - f(x)}{h}$

is not defined for for $$h = 0$$, but this is where limits come in handy. We can approximate the slope of the tangent line by taking the limit of the above expression as $$h \to 0$$ (if the limit exists). We call this limit the derivative of the function $$f(x)$$, and denote it $$f'(x)$$ or $$\dfrac{d}{dx}\left(f(x)\right)$$.

The point $$P$$ is $$(x, f(x))$$, and the point $$Q$$ is $$(x+h, f(x+h))$$. As point $$Q$$ moves closer to point $$P$$, the distance between the two points, $$h$$, also decreases.

Observe that as this happens, the approximated tangent, represented by the blue dashed line, approaches the actual tangent at point $$P$$, represented by the red line.

Try changing the function, $$f(x)$$, in the left side panel, and play around by moving points $$P$$ and $$Q$$.

Tip: You can drag the point $$N$$ horizontally.