How do you find the first derivative of a function?

The first derivative of a function can be found using the limit definition of the derivative. According to the reference provided, the process involves several steps:

Steps to Finding the First Derivative:

1. Substitute (x+h) for x: Replace every instance of 'x' in the original function with '(x+h)'.
2. Subtract the Original Function: Subtract the original function, f(x), from the modified function, f(x+h). This gives you f(x+h) - f(x).
3. Divide by h: Divide the entire expression [f(x+h) - f(x)] by 'h'. This yields [f(x+h) - f(x)] / h.
4. Simplify the Expression: Use your algebra skills to simplify the resulting expression. This often involves expanding terms, canceling common factors, and combining like terms.
5. Take the Limit as h Approaches 0: (This isn't explicitly mentioned in the provided reference, but it is a crucial step in finding the derivative). Determine what happens to the simplified expression as 'h' gets infinitely close to zero. This final expression represents the first derivative of the function.

In mathematical notation, the first derivative, denoted as f'(x) or dy/dx, is defined as:

f'(x) = lim (h->0) [f(x+h) - f(x)] / h

Example:

Let's say we want to find the first derivative of f(x) = x².

1. Substitute (x+h) for x: f(x+h) = (x+h)² = x² + 2xh + h²
2. Subtract the Original Function: f(x+h) - f(x) = (x² + 2xh + h²) - x² = 2xh + h²
3. Divide by h: [f(x+h) - f(x)] / h = (2xh + h²) / h = 2x + h
4. Simplify the Expression: 2x + h is already simplified.
5. Take the Limit as h Approaches 0: lim (h->0) (2x + h) = 2x

Therefore, the first derivative of f(x) = x² is f'(x) = 2x.