Problem 8 - Calculus an Applied Approach

Question

find the second derivative of the function. $$ g(t)=32 t^{-2} $$

Step-by-Step Solution

Answer

The second derivative of the function \( g(t) = 32t^{-2} \) is \( g''(t) = 192t^{-4} \).

Step 1: Find the first derivative

According to the power rule, the first derivative of the function \( g(t)=32t^{-2} \) will be computed as follows: \( g'(t)= -2(32)t^{-3}= -64t^{-3} \).

Step 2: Find the second derivative

To find the second derivative, apply the power rule again, but this time to the first derivative: \( g''(t)= -3(-64)t^{-4}= 192t^{-4} \).

Key Concepts

Power Rule Differentiation

The power rule is a fundamental technique in calculus for finding the derivative of a function that is a power of a variable. It states that if you have a function of the form \( f(x) = ax^n \), where \( a \) is a constant and \( n \) is a real number, the derivative of that function with respect to \( x \) is \( f'(x) = anx^{n-1} \).

Let's consider the exercise where we have the function \( g(t) = 32t^{-2} \). Using the power rule differentiation, we find the first derivative by reducing the power of \( t \) by 1 and multiplying the coefficient by the original power, resulting in \( g'(t) = -2 \times 32t^{-2-1} = -64t^{-3} \).

This rule makes computing derivatives much faster and is particularly useful when dealing with polynomial functions where each term can be differentiated independently.

Derivative of Inverse Functions

When finding the derivative of inverse functions, it’s essential to understand the relationship between a function and its inverse. If \( f(x) \) is a one-to-one function with an inverse \( f^{-1}(y) \), then the derivative of the inverse function at a point \( y \) is the reciprocal of the derivative of the function at the point \( x \), such that \( f(x) = y \).

Mathematically, this can be expressed as \( (f^{-1})'(y) = \frac{1}{f'(x)} \), given that \( f'(x) \) is not equal to zero. This concept isn't directly applied in our exercise since we are dealing with the power function rather than an inverse function. However, understanding how to differentiate inverse functions is crucial for solving a wide range of calculus problems where functions are defined implicitly or have inverse operations.

Calculus Problem Solving

Calculus problem solving involves a step-by-step approach to find derivatives, integrals, limits, and more. To successfully solve calculus problems, one must understand the application of rules and theorems, pay close attention to detail, and practice various types of problems to become proficient.

In the example provided, we applied the power rule for differentiation not once, but twice to find the second derivative of the function \( g(t) = 32t^{-2} \). After obtaining the first derivative, \( g'(t) = -64t^{-3} \), we applied the power rule again to find the second derivative \( g''(t) \), arriving at \( g''(t) = 192t^{-4} \). This illustrates a systematic approach to tackle calculus problems: understanding the given function, identifying the right rule to apply, and carrying out the differentiation process step by step to reach the solution.