# Differentiation from First Principles

## Understanding Differentiation from First Principles

Differentiation is a fundamental process used in mathematics and physics to analyze the change of a variable function. Also known as finding the gradient, slope, or rate of change, it involves using standard straight line graph techniques to calculate how a function changes over time. In this article, we will take a closer look at differentiation from first principles, a method that uses these techniques to determine the gradient of a function.

To begin, let's familiarize ourselves with the formula behind differentiation from first principles. Suppose we have a function, y = f(x), and we want to find its gradient at a specific point. We must select two points, x and x + h, and determine their coordinates, which we will call (x, f(x)) and (x + h, f(x + h)).

Next, we can utilize the formula **limh->0((f(x+h)-f(x))/h)** to calculate the gradient. This formula, often found in formula booklets, is an essential concept in differentiation from first principles. Let's walk through an example to see this in action.

## Worked Examples of Differentiation from First Principles

Let's begin with a simple example: differentiate from first principles.

**Step 1:** Take y = x^2 as our function. We can choose two points, x and x + h, and determine their coordinates to be (x, x^2) and (x + h, (x + h)^2).

**Step 2:** Using the formula, we calculate the difference quotient to be ((x+h)^2 - x^2)/h.

**Step 3:** Simplifying this expression, we get (2x + h).

**Step 4:** Finally, by taking the limit of this expression as h approaches 0, we find the derivative of y = x^2, which is 2x. And just like that, we have successfully differentiated a quadratic function from first principles!

But what about more complex functions? Let's try differentiating from first principles.

**Step 1:** Consider y = e^x as our function. We can choose two points, x and x + h, and determine their coordinates to be (x, e^x) and (x + h, e^(x + h)).

**Step 2:** Using the formula, we calculate the difference quotient to be ((e^(x+h)) - e^x)/h.

**Step 3:** Simplifying this expression, we get (e^x)(e^h - 1)/h.

**Step 4:** Taking the limit as h approaches 0, we are left with the derivative of y = e^x, which is e^x.

We can see that differentiation from first principles involves taking the limit of the gradient between two points of a function. And as shown by these examples, the final answers align with the results obtained using traditional differentiation methods.

Now, let's try a more intricate example utilizing an exponential function: differentiate from first principles.

**Step 1:** Let y = f(x) be our function. We can choose two points, x and x + h, and find their coordinates to be (x, f(x)) and (x + h, f(x + h)).

**Step 2:** By applying the formula, we calculate the difference quotient to be ((f(x+h)) - f(x))/h.

**Step 3:** Simplifying this expression, we get (f'(x) + hf''(x) + ...) + O(h^2).

**Step 4:** Taking the limit as h approaches 0, we obtain the derivative of y = f(x), which is f'(x). This may seem complicated, but it essentially involves finding the derivative of a linear function.

## Key Takeaways from Differentiation from First Principles

- Differentiation is the process of finding the gradient of a curve.
- Using differentiation from first principles, we can determine the gradient of a curve at any given point.
- The formula for differentiation from first principles can be found in formula booklets and is a fundamental concept in calculus.
- By taking the limit as h approaches 0, we can find the derivative of a function using differentiation from first principles.