21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 4x²+1; a= 2,4

Verified step by step guidance

1. Start by recalling the definition of the derivative using limits: f'(x) = lim(h→0) [(f(x+h) - f(x))/h].
2. Substitute the given function f(x) = 4x² + 1 into the derivative definition: f'(x) = lim(h→0) [(4(x+h)² + 1 - (4x² + 1))/h].
3. Expand the expression (x+h)² to get x² + 2xh + h², and substitute it back into the limit: f'(x) = lim(h→0) [(4(x² + 2xh + h²) + 1 - 4x² - 1)/h].
4. Simplify the expression inside the limit: f'(x) = lim(h→0) [(4x² + 8xh + 4h² + 1 - 4x² - 1)/h] = lim(h→0) [(8xh + 4h²)/h].
5. Factor out h from the numerator and cancel it with the denominator: f'(x) = lim(h→0) [8x + 4h]. As h approaches 0, the term 4h vanishes, leaving f'(x) = 8x.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In mathematical terms, the derivative f'(x) is given by the limit: f'(x) = lim(h→0) [f(x+h) - f(x)] / h.

Limit

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It is essential for defining derivatives and integrals. The limit allows us to analyze the function's behavior at points where it may not be explicitly defined, such as at points of discontinuity or at infinity.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of finding derivatives, evaluating the function at points a=2 and a=4 is necessary to compute the derivative using the limit definition. This process helps in understanding how the function behaves at those specific points.