The derivative of f(x) is the mathematical function that describes the rate of change of f(x) with respect to x. In other words, it tells us how much f(x) changes for a small change in x. In this case, the derivative of f(x) = 1/2x^2 + 3x^3 is f'(x) = x + 9x^2.
To find the derivative of a function, we use the rules of differentiation, which involve taking the limit of a difference quotient. In simpler terms, we take the derivative of each term in the function and combine them using the sum, difference, and product rules.
The derivative of a function is important because it helps us understand the behavior of the function. It tells us the slope of the function at any given point, which can help us find critical points, determine maximum and minimum values, and solve optimization problems.
The chain rule is a rule of differentiation that allows us to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In other words, we take the derivative of the outer function and replace the inner function with its derivative.
The derivative can be used to graph a function by helping us identify important points such as local extrema, points of inflection, and intervals of increase or decrease. By plotting these points and connecting them with smooth curves, we can create an accurate graph of the original function.