21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(t) = 1/√t; a=9, 1/4

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the derivative of the function f(t) = \frac{1}{\sqrt{t}} using the definition of the derivative, which involves limits.
2. Step 2: Recall the definition of the derivative. The derivative f'(t) is given by the limit: f'(t) = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}.
3. Step 3: Substitute the function into the definition. For f(t) = \frac{1}{\sqrt{t}}, we have f(t+h) = \frac{1}{\sqrt{t+h}}. Substitute these into the limit: f'(t) = \lim_{h \to 0} \frac{\frac{1}{\sqrt{t+h}} - \frac{1}{\sqrt{t}}}{h}.
4. Step 4: Simplify the expression. To simplify the expression, find a common denominator for the terms in the numerator: \frac{1}{\sqrt{t+h}} - \frac{1}{\sqrt{t}} = \frac{\sqrt{t} - \sqrt{t+h}}{\sqrt{t} \cdot \sqrt{t+h}}.
5. Step 5: Rationalize the numerator. Multiply the numerator and the denominator by the conjugate of the numerator, \sqrt{t} + \sqrt{t+h}, to eliminate the square roots in the numerator. This will help in simplifying the limit expression.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In calculus, the derivative is often denoted as f'(t) and can be interpreted as the slope of the tangent line to the function's graph at a given point.

Limits

Limits are fundamental in calculus, used to define both derivatives and integrals. A limit describes the behavior of a function as its input approaches a certain value. In the context of derivatives, the limit is used to find the instantaneous rate of change by evaluating the function's behavior as the interval between two points shrinks to zero.

Function Notation

Function notation is a way to represent mathematical functions in a clear and concise manner. In this case, f(t) = 1/√t indicates that f is a function of t, where the output is the reciprocal of the square root of t. Understanding function notation is essential for applying calculus concepts, such as finding derivatives, as it allows for precise communication of mathematical relationships.