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.
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.
