Introduction to Derivatives

Information technology is all about slope!

Gradient = Change in Y Change in X

gradient

We can discover an average slope between two points.

 average slope = 24/15

But how do we find the slope at a signal?

There is nothing to measure!

 slope 0/0 = ????

But with derivatives we use a small deviation ...

... then have it shrink towards null.

 slope delta y / delta x

Let us Discover a Derivative!

To discover the derivative of a role y = f(x) we apply the slope formula:

Slope = Change in Y Change in Ten = Δy Δx

slope delta x and delta y

And (from the diagram) we see that:

10 changes from 10 to 10+Δx
y changes from f(ten) to f(10+Δx)

Now follow these steps:

  • Make full in this slope formula: Δy Δx = f(x+Δx) − f(x) Δx
  • Simplify it as all-time we tin
  • Then brand Δx compress towards zero.

Similar this:

Instance: the office f(10) = xtwo

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

Start with: f(x+Δx) = (x+Δx)2
Expand (ten + Δx)2: f(x+Δx) = x2 + 2x Δx + (Δx)2

The slope formula is: f(x+Δx) − f(10) Δx

Put in f(x+Δx) and f(x): ten2 + 2x Δx + (Δx)two − 102 Δx

Simplify (x2 and −xii abolish): 2x Δx + (Δx)2 Δx

Simplify more (divide through by Δx): = 2x + Δx

Then, equally Δx heads towards 0 we get: = 2x

Consequence: the derivative of x2 is 2x

In other words, the gradient at x is 2x

We write dx instead of "Δx heads towards 0".

And "the derivative of" is usually written d dx similar this:

d dx tentwo = 2x
"The derivative of tentwo equals 2x"
or merely "d dx of 102 equals 2x"


And so what does d dx 102 = 2x mean?

slope x^2 at 2 is 4

Information technology means that, for the function tenii, the gradient or "rate of change" at any signal is 2x.

So when 10=2 the gradient is 2x = iv, equally shown hither:

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

Note: f'(10) can likewise be used for "the derivative of":

f'(ten) = 2x
"The derivative of f(x) equals 2x"
or simply "f-dash of ten equals 2x"

Permit's try another case.

Example: What is d dx tenthree ?

We know f(10) = teniii , and can summate f(x+Δx) :

Kickoff with: f(x+Δx) = (x+Δx)three
Expand (x + Δx)3: f(x+Δx) = xiii + 3x2 Δx + 3x (Δx)two + (Δx)3

The slope formula: f(ten+Δx) − f(x) Δx

Put in f(10+Δx) and f(x): x3 + 3xtwo Δx + 3x (Δx)2 + (Δx)3 − xiii Δx

Simplify (x3 and −xiii cancel): 3x2 Δx + 3x (Δx)2 + (Δx)3 Δx

Simplify more (split up through by Δx): 3xii + 3x Δx + (Δx)2

And so, equally Δx heads towards 0 nosotros go: 3x2

Result: the derivative of xiii is 3x2

Have a play with information technology using the Derivative Plotter.

Derivatives of Other Functions

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

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

On Derivative Rules it is listed every bit being cos(x)

Washed.

But using the rules can exist tricky!

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

We get a incorrect reply if we endeavour 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 folio.

And information technology actually works out to be costwo(ten) − sin2(x)

So that is your adjacent step: learn how to use the rules.

Notation

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

f'(x) = lim Δx→0 f(ten+Δx) − f(x) Δx

"The derivative of f equals
the limit equally 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):

dy dx = f(10+dx) − f(ten) dx

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

Yous do differentiation ... to get a derivative.

Where to Side by side?

Go and larn how to find derivatives using Derivative Rules, and get plenty of do: