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Differentiation from First Principles

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


Frequently Asked Questions about Differentiation from First Principles

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.

Show answer

Answer

3

Show question

Question

Differentiate from first principles 5x.

Show answer

Answer

5

Show question

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