View: https://www.youtube.com/watch?v=BcOPKQAZcn0

Introduction to Derivatives​

It is all about slope!

Let us Find a Derivative!​

To find the derivative of a function y = f(x) we use the slope formula:

Slope = Change in YChange in X = ΔyΔx



And (from the diagram) we see that:

Now follow these steps:

• Fill in this slope formula: ΔyΔx = f(x+Δx) − f(x)Δx
• Simplify it as best we can
• Then make Δx shrink towards zero.

Like this:

Example: the function f(x) = x2​

We know f(x) = x2, and we can calculate f(x+Δx) :

The slope formula is:f(x+Δx) − f(x)Δx
Put in f(x+Δx) and f(x):x2 + 2x Δx + (Δx)2 − x2Δx
Simplify (x2 and −x2 cancel):2x Δx + (Δx)2Δx
Simplify more (divide through by Δx):= 2x + Δx
Then, as Δx heads towards 0 we get:= 2x

Result: the derivative of x2 is 2x
In other words, the slope at x is 2x


We write dx instead of "Δx heads towards 0".
And "the derivative of" is commonly written ddx like this:
ddxx2 = 2x
"The derivative of x2 equals 2x"
or simply "d dx of x2 equals 2x"

So what does ddxx2 = 2x mean?​

It means that, for the function x2, the slope or "rate of change" at any point is 2x.

So when x=2 the slope is 2x = 4, as shown here:

Or when x=5 the slope is 2x = 10, and so on.

Note: f’(x) can also be used for "the derivative of":
f’(x) = 2x
"The derivative of f(x) equals 2x"
or simply "f-dash of x equals 2x"


Let's try another example.

Example: What is ddxx3 ?​

We know f(x) = x3, and can calculate f(x+Δx) :

The slope formula:f(x+Δx) − f(x)Δx
Put in f(x+Δx) and f(x):x3 + 3x2 Δx + 3x (Δx)2 + (Δx)3 − x3Δx
Simplify (x3 and −x3 cancel):3x2 Δx + 3x (Δx)2 + (Δx)3Δx
Simplify more (divide through by Δx):3x2 + 3x Δx + (Δx)2
Then, as Δx heads towards 0 we get:3x2
Result: the derivative of x3 is 3x2
Have a play with it using the Derivative Plotter.

Derivatives of Other Functions​

We can use the same method to work out derivatives of other functions (like sine, cosine, logarithms, etc).

But in practice the usual way to find derivatives is to use:
Derivative Rules

Example: what is the derivative of sin(x) ?​

On Derivative Rules it is listed as being cos(x)
Done.
But using the rules can be tricky!

Example: what is the derivative of cos(x)sin(x) ?​

We get a wrong answer if we try to multiply the derivative of cos(x) by the derivative of sin(x) ... !
Instead we use the "Product Rule" as explained on the Derivative Rules page.
And it actually works out to be cos2(x) − sin2(x)
So that is your next step: learn how to use the rules.

Notation​

"Shrink towards zero" is actually written as a limit like this:

f’(x) = limΔx→0 f(x+Δx) − f(x)Δx
"The derivative of f equals
the limit as Δx goes to zero of f(x+Δx) - f(x) over Δx"



Or sometimes the derivative is written like this (explained on Derivatives as dy/dx):

dydx = f(x+dx) − f(x)dx


The process of finding a derivative is called "differentiation".