Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(f^{\prime}(2)\) if \(f(t)=(2 t)^{2}\)

Step-by-Step Solution

Answer

The derivative is 16.

Step 1: Write down the function

The given function is \( f(t) = (2t)^2 \).

Step 2: Simplify the function

Simplify the function by expanding the square: \( f(t) = 4t^2 \).

Step 3: Find the derivative of the function

To find the derivative \( f'(t) \), use the power rule \( \frac{d}{dt}[t^n] = nt^{n-1} \):\[ f'(t) = \frac{d}{dt}[4t^2] = 8t \]

Step 4: Evaluate the derivative at \( t = 2 \)

Substitute \( t = 2 \) into the derivative:\[ f'(2) = 8 \times 2 = 16 \]

Key Concepts

Power Rule

When dealing with derivatives, one of the most fundamental rules is the **Power Rule**. It is a straightforward technique that allows us to find derivatives of functions of the form \( x^n \). If you have a function \( f(x) = x^n \), the Power Rule states that the derivative \( f'(x) \) can be easily calculated as \( nx^{n-1} \).
For example, when you apply this to \( f(t) = 4t^2 \), the power of \( t \) is 2. Thus, using the power rule, bring down the 2 (the exponent) and multiply it by 4 (the coefficient of \( t^2 \)), to get \( 8t \). This new power of \( t \) becomes \( t^{2-1} = t^1 \), which simplifies to \( t \).
So, the derivative by the Power Rule transforms our function from \( 4t^2 \) to \( 8t \). Remember, the Power Rule is highly efficient and must always be applied liberally to polynomials where terms are in \( x^n \) format.

Evaluating Derivatives

After finding the derivative using the power rule or any other method, the next step often involves evaluating this derivative at a particular point. This means you need to substitute a specific value into the derived formula.

In this exercise, we calculated \( f'(t) = 8t \) from the simplified function. Now, we need to evaluate this expression at \( t = 2 \). This process involves:

• Replacing \( t \) in \( 8t \) with the value 2.
• Carrying out the multiplication: \( 8 \times 2 \ = 16 \).

The result, \( f'(2) = 16 \), tells us the rate of change of the original function at \( t = 2 \).
Evaluating derivatives at specific points gives crucial insights into the behavior of the function at those points, notably how quickly it increases or decreases.

Function Simplification

Function simplification is about making the function easier to work with before finding its derivative. This initial step is essential because it can save a lot of time and effort when applying derivative rules.

In the example given, the function \( f(t) = (2t)^2 \) looks complex at first. By expanding it, you simplify the function to \( f(t) = 4t^2 \). This reformation makes it straightforward to then apply the Power Rule.

• Expansion involves multiplying out terms, in this case from \( (2t)^2 \) to \( 4t^2 \).
• Simplifying often means reducing a function to its most basic polynomial form.

The simplification process can be as simple as distributing powers across products. These steps are crucial as they simplify the algebra and allow you to focus directly on applying derivative rules effectively.
Always look for opportunities to simplify before diving into further calculations. This makes finding derivatives, such as using the Power Rule, much more straightforward and manageable.