Finding Derivative Functions and Values 


Using the definition, calculate the derivatives of the functions in Exercises 1–6. Then find the values of the derivatives as specified.


p(θ) = √3θ; p′(1), p′(3), p′(2/3)

Verified step by step guidance

1. Start by recalling the definition of the derivative: the derivative of a function p(θ) at a point θ is given by the limit as h approaches 0 of [p(θ + h) - p(θ)] / h.
2. For the function p(θ) = √3θ, substitute into the definition: p'(θ) = lim(h→0) [(√3(θ + h) - √3θ) / h].
3. Simplify the expression inside the limit: √3(θ + h) = √3θ + √3h. Therefore, the expression becomes lim(h→0) [(√3θ + √3h - √3θ) / h] = lim(h→0) [√3h / h].
4. Cancel h in the numerator and denominator: lim(h→0) [√3h / h] = lim(h→0) [√3]. Since √3 is a constant, the limit is simply √3.
5. Now, evaluate the derivative at the specified points: p'(1) = √3, p'(3) = √3, and p'(2/3) = √3, since the derivative is constant for all θ.

Key Concepts

Definition of Derivative

The derivative of a function at a point is the limit of the average rate of change of the function as the interval approaches zero. Mathematically, it is defined as f'(x) = lim(h→0) [f(x+h) - f(x)]/h. This concept is crucial for understanding how to calculate the instantaneous rate of change of a function.

Power Rule for Derivatives

The power rule is a basic derivative rule used to find the derivative of functions in the form of f(x) = x^n. It states that the derivative of x^n is n*x^(n-1). This rule simplifies the process of finding derivatives for polynomial functions and is essential for calculating derivatives efficiently.

Substitution in Derivatives

Substitution involves replacing the variable in the derivative with specific values to find the derivative at those points. For example, after finding the derivative p'(θ), substitute θ with 1, 3, and 2/3 to find p'(1), p'(3), and p'(2/3). This step is necessary to evaluate the derivative at given points, providing specific rates of change.