Suggested languages for you:
|
|

## All-in-one learning app

• Flashcards
• NotesNotes
• ExplanationsExplanations
• Study Planner
• Textbook solutions

# Differentiation from First Principles

Differentiation is the process of finding the gradient of a variable function. A variable function is a polynomial function that takes the shape of a curve, so is therefore a function that has an always-changing gradient.

There is a traditional method to differentiate functions, however, we will be concentrating on finding the gradient still through differentiation but from first principles.

This means using standard Straight Line Graphs methods of to find the gradient of a function.

## How do we differentiate from first principles?

This involves using to calculate the gradient of a function.

STEP 1: Let be a function. Pick two points and .

The coordinates of x will be and the coordinates of will be.

STEP 2: Find and .

STEP 3: Complete .STEP 4: Take a limit..

The formula above is often found in the formula booklets that are given to students.

Let’s look at an example to put this all together.

## Worked examples of differentiation from first principles

Let's look at two examples, one easy and one a little more difficult.

Differentiate from first principles .

SOLUTION:

 STEP Example STEP 1: Let be a function. Pick two points and . Coordinates are and .We can simplify STEP 2: Find and . STEP 3:Complete STEP 4: Take a limit. ANSWER however the entire proof is a differentiation from first principles.

Let’s look at a more complicated example.

Differentiate from first principles .

SOLUTION:

 STEP Example STEP 1: Let y = f(x) be a function. Pick two points x and x + h. Coordinates are and.We can use addition formulae to simplify STEP 2: Find and STEP 3:Complete STEP 4: Take a limit. ANSWER however the entire proof is a differentiation from first principles.

So differentiation can be seen as taking a limit of a gradient between two points of a function. You will see that these final answers are the same as taking derivatives.

Let's look at another example to try and really understand the concept. This time we are using an exponential function.

Differentiate from first principles .

SOLUTION:

 STEP 1: Let y = f(x) be a function. Pick two points x and x + h. Co-ordinates are and . STEP 2: Find and STEP 3:Complete STEP 4: Take a limit. ANSWER , but of course, the entire proof is an answer as this is differentiation from first principles.

## Differentiation from First Principles - Key takeaways

• Differentiation is the process of finding the gradient of a curve.
• The gradient of a curve changes at all points.
• Differentiation can be treated as a limit tending to zero.
• The formula to differentiate from first principles is found in the formula booklet and is

We take the gradient of a function using any two points on the function (normally x and x+h).

The formula is:

limh->0((f(x+h)-f(x))/h)

We simply use the formula and cancel out an h from the numerator. This should leave us with a linear function.

We use addition formulae to simplify the numerator of the formula and any identities to help us find out what happens to the function when h tends to 0.

## Final Differentiation from First Principles Quiz

Question

Differentiate from first principles 3x.

3

Show question

Question

Differentiate from first principles 5x.

5

Show question

More about Differentiation from First Principles
60%

of the users don't pass the Differentiation from First Principles quiz! Will you pass the quiz?

Start Quiz

## Study Plan

Be perfectly prepared on time with an individual plan.

## Quizzes

Test your knowledge with gamified quizzes.

## Flashcards

Create and find flashcards in record time.

## Notes

Create beautiful notes faster than ever before.

## Study Sets

Have all your study materials in one place.

## Documents

Upload unlimited documents and save them online.

## Study Analytics

Identify your study strength and weaknesses.

## Weekly Goals

Set individual study goals and earn points reaching them.

## Smart Reminders

Stop procrastinating with our study reminders.

## Rewards

Earn points, unlock badges and level up while studying.

## Magic Marker

Create flashcards in notes completely automatically.

## Smart Formatting

Create the most beautiful study materials using our templates.