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Find the derivative of the function using the definition of...

Key Concepts

Domain of its Derivative

The domain of a derivative usually coincides with the portion of the domain of the original function where the function is differentiable. In the case of smooth functions like polynomials, the derivative exists for all values in the original domain, meaning that the domain of the derivative is the same as that of the function.

Polynomial Functions

Polynomial functions are algebraic expressions involving sums of powers of x with constant coefficients, and they are smooth and continuous everywhere in their domain. Because of their simple structure, polynomial functions are differentiable at every point on the real number line, and their derivatives are also polynomial functions.

Domain of a Function

The domain of a function is the complete set of input values for which the function is defined. For many functions, especially polynomials, the domain is typically the set of all real numbers. Understanding the domain is crucial as it defines the scope within which a function can be evaluated.

Definition of Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the increment approaches zero. This concept formalizes the idea of instantaneous rate of change, and it is expressed as f'(x) = lim (h ? 0) [(f(x+h) - f(x))/h]. It is a fundamental concept in calculus that links the notions of limits and instantaneous change.

Limit Process

Using the limit process means computing the derivative directly from the definition, rather than applying shortcut differentiation rules. This involves carefully expanding the function at x + h, simplifying the difference quotient, and taking the limit as h tends to zero. Understanding this process is important as it reinforces the meaning of the derivative and the role of limits in calculus.

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