Problem 29 - Calculus - AP Edition

Question

Find the indicated derivative. $$\frac{d}{d t}\left[16 t^{2}\right]$$

Step-by-Step Solution

Answer

The derivative of \(16t^2\) with respect to \(t\) is \(32t\).

Step 1: Identify the Function and the Variable

You are given the function \(16t^2\) and need to find its derivative with respect to the variable \(t\). This means we want to apply derivative rules to \(16t^2\) to find \(\frac{d}{dt}[16t^2]\).

Step 2: Use the Power Rule for Derivatives

The power rule states that the derivative of \(t^n\) with respect to \(t\) is \(nt^{n-1}\). Here, we can apply this rule to find \(\frac{d}{dt}[t^2]\), which is \(2t\).

Step 3: Apply Constant Multiplication Rule

When a function is multiplied by a constant, the derivative of the constant times a function is the constant times the derivative of the function. For \(16t^2\), you multiply the constant 16 by the derivative of \(t^2\), which we found to be \(2t\). Therefore, \(\frac{d}{dt}[16t^2] = 16 \cdot 2t\).

Step 4: Simplify the Expression

Now simplify the expression \(16 \cdot 2t\) to get the final derivative. Multiplying 16 by 2 gives 32, so we have \(32t\).

Key Concepts

The Power Rule in Calculus

When dealing with derivatives, the power rule is a fundamental tool. It simplifies finding the derivative of polynomial functions. If you have a function of the form \( t^n \), where \( n \) is a real number, the power rule states that the derivative is \( nt^{n-1} \). This means you multiply the exponent \( n \) by the base \( t \) raised to the one-less power.
For example, if \( n = 2 \), as in \( t^2 \), the derivative is \( 2t^{2-1} = 2t \).

• Start by identifying the exponent in the term.
• Multiply the exponent by the term.
• Decrease the exponent by 1 to find the new power of the variable.

This process is quick and very useful for functions with t raised to any power. Once you understand this rule, derivatives become much easier to manage, especially for polynomial expressions.

Understanding the Constant Multiplication Rule

The constant multiplication rule assists when you need to differentiate a function multiplied by a constant number. A constant is a fixed number that doesn't change; in our case, the function is \(16t^2\). This rule states that to find the derivative of a constant multiplied by a function, you simply multiply the constant by the derivative of the function.
Let's break it down:

• Identify the constant, which is 16 in \(16t^2\).
• Find the derivative of the function without the constant. For \( t^2 \), we calculated it as \( 2t \) using the power rule.
• Multiply the constant by the derivative of the function: \( 16 \times 2t \).

This gives you the final derivative, \( 32t \). This rule helps simplify complex derivatives quickly and efficiently by isolating constants early in the calculation.

Basics of Differentiation

Differentiation is a core concept in calculus and involves calculating the rate at which a function changes. It's essentially finding the derivative of a function. This process requires certain rules, like the power rule and constant multiplication rule, to make the task easier.
To perform differentiation:

• Identify the variable you're differentiating with respect to. In \(16t^2\), the variable is \(t\).
• Apply applicable derivative rules to each part of the function. For polynomial expressions, the power rule and constant multiplication rule are very handy.
• Simplify the expression to get the final derivative. For our function, the differentiation process concluded with \( 32t \).

Understanding differentiation allows you to analyze how functions behave, which is critical in fields like physics, economics, and beyond. It's an essential skill for evaluating how quantities change in real-world scenarios.