The first derivative represents the rate of change of a function at a given point. It measures how much a function's output changes with respect to its input.
The first derivative is calculated using the derivative formula, which involves taking the limit of a function as its input approaches a specific value. This is also known as the difference quotient.
The chain rule is a method for finding the derivative of a composite function, which is a function that is composed of two or more other functions. It allows us to break down the derivative of a composite function into smaller, more manageable parts.
The chain rule is important because it allows us to find the derivative of complicated functions by breaking them down into simpler parts. It is a fundamental tool in calculus and is used in many real-world applications, such as in physics, engineering, and economics.
Sure, let's say we have the function f(x) = (x^2 + 1)^3. Using the chain rule, we can break this down into two simpler functions: g(x) = x^2 + 1 and h(x) = x^3. Then, the derivative of f(x) can be found by multiplying the derivative of g(x) (2x) with the derivative of h(x) (3x^2), resulting in f'(x) = 6x(x^2 + 1)^2.