Derivative by definition


How do you find slope if you only know 1 point ???

In the prior lesson on "what is a derivative", we spent a lot of time trying to understand the difference between a secant and a tangent and, at the end, we learned that a derivative is the slope of a line at a specific point. However, there is a major problem here:

A Specific Example of our dilemma

Imagine that we have some curve, like the red one below, and we know that the point (1,2) is on that curve. The slope of the purple line is the derivative so all that we need to know is the slope of the purple line.

But how do we find the slope of a line when we only know 1 point ?
$ slope = \frac{\Delta y}{\Delta x} = \frac{2 - \color{red}{?} }{1 - \color{red}{?} } $

How do we define the tangent mathematically?

There are two common definitions of tangent. Both of these definitions are equivalent definitions of the derivative at a point. They are presented side by side below.

One of the notations used to represent the derivative is $ f' $

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Version 1
version 1 definition

Version 2

Definition picture 2

This version represents the derivative with respect to the point
$$ (a,f(a)) , (\text{i.e. } f'(a) ) \\ f'(a) = \lim_{x\rightarrow a} \frac{ f(x) - f(a)}{ x-a} $$

What does this mean?

The formula above means
Find the slope between the points (x,f(x)) and (a,f(a)) such that the point (x,f(x)) gets infinitesimally close to (a,f(a))
In this equation, we are saying find the slope of the tangent line between two points when you add almost zero to go from the first $ x_0 $ value to the next.

$$f'(x_0) = \lim_{h\rightarrow 0} = \frac{ f(x_0 + h) - f(x_0)}{ h } $$ What does this mean?

This formula represents the same idea as version 1. We are saying that the points are practically identical because we only moved an extremely small amount over. Point one $(x_0, f(x_0)) $ and point two$(x_0 +h, f(x_0 +h)) $ so the $ \Delta x = (x_0 +h) - x_0 = h $

Example Problem 1

Find the slope of the tangent line of $y = x^2 + 3x $ using both formulas at x = 5

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Example Problem 2

Find the slope of the tangent line of $y = \sqrt{4x+1}$ using both formulas at x = 2

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Example Problem 3

Find the slope of the tangent line of $y = \frac{1}{x-1}$ using both formulas at x = 3

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Version 2

General Formula of the Derivative

The following definition is an extension of version 2 for any general point 'x' (instead of the specific value $ x_0 $ ).

The derivative with respect to x

$f'(x) = \lim_{h\rightarrow 0} = \frac{ f(x + h) - f(x)}{ h } $
This formula tells us the derivative of the function f(x), we can then use the answer to find teh slope at any point without have to keep repeating the formula .