Understanding Derivatives: the Basics

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Understanding Slope

Given two points on a graph a and b, we can determine the slope between the two points via:

$Slope = \frac{change\:in\:y}{change\:in\:x} = \frac{\triangle y}{\triangle x}$

But what if we want to find the slope at an individual point, instead of the difference between two? We can’t use change, because there is no change. This is where derivatives come in: we imagine a small difference, then reduce it to zero mathematically.

Calculating slope at a point

So, how do we determine the change in x? Well, we calculate y at x, and y at x + $\triangle x$ , and subtract the latter from the former. Specifically, it’s:

$\triangle y = f(x + \triangle x) - f(x)$

Superimposiing that into the slope equation above, we get:

$slope = \frac{f(x + \triangle x) - f(x)}{\triangle x}$

Then, after calculating the above, we reduce $\triangle x$ down to zero.

Let’s do an example:

$f(x) = x^2$

$f(x + \triangle x) = (x + \triangle x)^2$
$= (x + \triangle x)(x + \triangle x)$
$= x^2 + 2x\triangle x + (\triangle x)^2$

Putting that into the slope equation, we get:

$\frac{x^2 + 2x \triangle x + (\triangle x)^2 - x^2}{\triangle x}$
$= \frac{2x \triangle x + (\triangle x)^2}{\triangle x}$
$= 2x + \triangle x$

NOW, because we want the solution when change of x is zero, we can assume $\triangle x = 0$ . So therefore, the above becomes:

$2x + \triangle x$
$= 2x + 0$
$= 2x$

Put another way: the slope at x is 2x

Notation basics

"Derivative" notation is $\frac{d}{dx}$ . For example, we’d say:

$\frac{d}{dx}x^2 = 2x$

In words, that is: the derivative of $x^2$ equals 2x

Also: apostrophies can be used for stating "the derivative of". Eg:

$f'(x) = 2x$

So remind me: what does this equation mean?

It means the rate of change (aka slope) at any point in the above equation is 2x.

So if x = 5, the slope is 10
If x is 20, the slope is 40.

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