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.


g(t) = 1/t²; g′(−1), g′(2), g′(√3)

Verified step by step guidance

1. Start by recalling the definition of the derivative: the derivative of a function g(t) at a point t is given by the limit g'(t) = lim(h→0) [(g(t+h) - g(t))/h].
2. Substitute the given function g(t) = 1/t² into the definition of the derivative. This gives g'(t) = lim(h→0) [(1/(t+h)² - 1/t²)/h].
3. Simplify the expression inside the limit. Find a common denominator for the fractions in the numerator: (1/(t+h)² - 1/t²) = (t² - (t+h)²) / (t²(t+h)²).
4. Expand and simplify the numerator: t² - (t+h)² = t² - (t² + 2th + h²) = -2th - h². Substitute this back into the limit expression.
5. Evaluate the limit: g'(t) = lim(h→0) [(-2th - h²) / (h * t²(t+h)²)]. Cancel h from the numerator and denominator, then take the limit as h approaches 0 to find g'(t). Finally, substitute t = -1, t = 2, and t = √3 to find g'(-1), g'(2), and g'(√3) respectively.

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. According to the power rule, the derivative is f'(x) = n*x^(n-1). This rule simplifies the process of finding derivatives for polynomial functions, including rational functions like g(t) = 1/t².

Substitution in Derivatives

Once the derivative function is found, substitution involves plugging specific values into the derivative to find the rate of change at those points. For example, after finding g'(t), substitute t = -1, t = 2, and t = √3 to find g'(-1), g'(2), and g'(√3), respectively. This step is essential for evaluating the derivative at given points.