Definition of the Derivative

The video introduces the definition of the derivative as f'(x) = lim (h→0) [f(x+h) - f(x)] / h. This formula is fundamental for finding the derivative of a function using a limit process.

The first example demonstrates finding the derivative of f(x) = 5x - 4. It involves substituting f(x+h) and f(x) into the formula, simplifying the expression, canceling terms, and evaluating the limit to find that the derivative is 5.

This section explains how to find the derivative of f(x) = x^2. The process includes expanding (x+h)^2, simplifying the numerator, factoring out 'h', canceling 'h', and then applying direct substitution to get a derivative of 2x.

Here, the video tackles the derivative of f(x) = 1/x. This involves dealing with a complex fraction by multiplying the numerator and denominator by the common denominator. After simplification and cancellation of 'h', direct substitution yields the derivative -1/x^2.

This part illustrates finding the derivative of f(x) = sqrt(x). The key step here is to multiply the numerator and denominator by the conjugate of the numerator. After simplifying and canceling terms, the derivative is found to be 1 / (2*sqrt(x)).

This complex example combines the challenges of rational and radical functions. It requires multiplying by the common denominator first to eliminate complex fractions, and then multiplying by the conjugate of the numerator. Despite its length, the systematic cancellation and substitution lead to the derivative -4 / (x*sqrt(x)) or -4/x^(3/2).

The final example demonstrates finding the derivative of a polynomial function. It involves expanding (x+h)^2 and distributing terms, careful handling of negative signs, canceling like terms, factoring out 'h', and finally using direct substitution to arrive at the derivative 2x - 5.