Problem 29 - Calculus Single Variable

Question

Calculate the requested derivative. \(\ddot{g}(1)\) where \(g(t)=\left(3 t^{2}-2\right)^{-4}\)

Step-by-Step Solution

Answer

The second derivative \( \ddot{g}(1) = 696 \).

Step 1: Differentiate g(t) once

We have the function \( g(t) = (3t^2 - 2)^{-4} \). To find \( \ddot{g}(t) \), we need the second derivative. First, find the first derivative, \( \dot{g}(t) \), using the chain rule: \( \dot{g}(t) = -4(3t^2 - 2)^{-5} \cdot (6t) \). So, \( \dot{g}(t) = -24t(3t^2 - 2)^{-5} \).

Step 2: Differentiate g(t) twice

Now, find the second derivative \( \ddot{g}(t) \) by differentiating \( \dot{g}(t) = -24t(3t^2 - 2)^{-5} \) again. By applying the product rule and chain rule, we have \( \ddot{g}(t) = -24(3t^2 - 2)^{-5} - 24t \cdot (-5)(3t^2 - 2)^{-6} \cdot (6t) \). This simplifies to \( \ddot{g}(t) = -24(3t^2 - 2)^{-5} + 720t^2(3t^2 - 2)^{-6} \).

Step 3: Evaluate the second derivative at t=1

Now, substitute \( t = 1 \) into \( \ddot{g}(t) \): \( \ddot{g}(1) = -24(3 \cdot 1^2 - 2)^{-5} + 720 \cdot 1^2(3 \cdot 1^2 - 2)^{-6} \). This simplifies to \( \ddot{g}(1) = -24(1)^{-5} + 720(1)^{-6} \). Since \( (1)^{-n} = 1 \) for any \( n \), we have \( \ddot{g}(1) = -24 + 720 = 696 \).

Key Concepts

Understanding the Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of composite functions. A composite function is essentially a function nested inside another function. For example, for our exercise, we have \((3t^2 - 2)^{-4}\), where \( (3t^2 - 2) \) is inside \((-4)\) as an exponent. To differentiate it, we first take the derivative of the outer function, \( (3t^2 - 2)^{-4} \), with respect to the inner function, and then multiply by the derivative of the inner function itself.

• First, derivative of the outer: multiply by -4 and reduce the exponent by 1.
• Then, multiply by the derivative of the inner: differentiate \( 3t^2 - 2 \) to get \( 6t \).

Lastly, the chain rule is particularly useful because it simplifies the process of differentiating complex expressions. In practice, it helps break down a difficult derivative into more manageable parts.

Mastering the Product Rule

The product rule is essential when you need to differentiate expressions that are the product of two or more functions. This rule states that the derivative of a product of two functions, say \( u(t) \) and \( v(t) \), is given by \((u'v + uv')\). In our problem, we apply the product rule when taking the second derivative of \( \dot{g}(t) = -24t(3t^2 - 2)^{-5} \).

• Differentiating \( -24t \) yields \(-24\).
• The function \( (3t^2 - 2)^{-5} \) requires the chain rule again, followed by multiplication by \(-24t \).

By using both the chain and product rules effectively, we can achieve accurate and precise derivative calculations for complicated functions.

Calculus Problem Solving Techniques

When tackling calculus problems, especially when involving multiple differentiation rules such as chain and product rules, a systematic approach is crucial. Here are some strategies for effective calculus problem solving:

• Identify the structure: Determine whether the function involves basic differentiation, chain rule, product rule, or a combination.
• Execute systematically: Follow step-by-step to avoid skipping critical parts of the differentiation processes.
• Check results: Substitute values or simplify expressions to verify accuracy.

By practicing these techniques regularly, your problem-solving skills in calculus will become more efficient, leading to quicker and more accurate solutions.

Steps in Evaluating Derivatives

Evaluating derivatives is at the heart of calculus and often involves applying differentiation rules to determine the rate of change or slope of a function. For the exercise given, evaluating involves substituting a specific variable value into the derivative you've computed.

Here, to find \( dot{g}(1)\), after deriving \( \ddot{g}(t) = -24(3t^2 - 2)^{-5} + 720t^2(3t^2 - 2)^{-6} \), we substitute \( t = 1 \) to find its exact rate at that point:

• First calculate the components: \( (3 \cdot 1^2 - 2)^{-5} = 1\) and \( (3 \cdot 1^2 - 2)^{-6} = 1 \).
• Then simplify: \( -24 \cdot 1 + 720 \cdot 1 = 696 \).

This process of substitution and simplification is key to evaluating derivatives accurately at particular points.