Problem 13 - The Calculus with Analytic Geometry

Question

Differentiate the given function by applying the theorems of this section. $$ f(s)=\sqrt{3}\left(s^{3}-s^{2}\right) $$

Step-by-Step Solution

Answer

3 \sqrt{3} s^{2} - 2 \sqrt{3} s

Step 1: Apply the Constant Multiple Rule

Recognize that \( \sqrt{3} \) is a constant multiple. The derivative of a function multiplied by a constant is the constant multiplied by the derivative of the function. Hence \( f'(s) = \sqrt{3} \cdot \frac{d}{ds} \left( s^{3} - s^{2} \right) \).

Step 2: Use the Sum/Difference Rule

Apply the sum and difference rule of differentiation, which states that the derivative of a sum or difference of functions is the sum or difference of their derivatives. So, \(\frac{d}{ds} \left( s^{3} - s^{2} \right) = \frac{d}{ds} s^{3} - \frac{d}{ds} s^{2} \).

Step 3: Differentiate Each Term

Use the power rule for differentiation. The power rule states that for any function of the form \( x^n \), the derivative is \( nx^{n-1} \). So, \( \frac{d}{ds} s^{3} = 3s^{2} \) and \( \frac{d}{ds} s^{2} = 2s \). Therefore, \( \frac{d}{ds} \left( s^{3} - s^{2} \right) = 3s^{2} - 2s \).

Step 4: Combine the Results

Multiply the derivative obtained in step 3 by the constant \( \sqrt{3} \), as determined in step 1. Thus, \( f'(s) = \sqrt{3} \cdot (3s^{2} - 2s) \).

Step 5: Simplify

Distribute and simplify the expression to get the final answer: \( f'(s) = 3 \sqrt{3} s^{2} - 2 \sqrt{3} s \).

Key Concepts

Constant Multiple Rule

When differentiating a function that is multiplied by a constant, you can simplify your work by using the Constant Multiple Rule. This rule says that you can take the constant out of the differentiation process, differentiate the function, and then multiply the result by the constant.

For example, in the function \( f(s) = \sqrt{3}(s^3 - s^2) \), the \( \sqrt{3} \) is a constant.

• First, we recognize that the constant doesn't change when we differentiate.
• Next, we differentiate the function inside the parentheses.

Our function becomes: \( f'(s) = \sqrt{3} \frac{d}{ds}(s^3 - s^2) \).

This step helps to simplify the differentiation process, making it easier to handle more complicated functions.

Sum/Difference Rule

Differentiating sums or differences of functions is straightforward with the Sum/Difference Rule. This rule states that the derivative of a sum or difference is simply the sum or difference of the derivatives.

For our function, we apply this rule to differentiate the terms inside the parentheses separately.

• The expression \( \frac{d}{ds}(s^3 - s^2) \) can be split into \( \frac{d}{ds}s^3 - \frac{d}{ds}s^2 \).
• This way, we handle each term individually.

Using this rule ensures that we don't miss any part of the function and makes the process clearer. Summing or subtracting the derivatives of simpler functions helps us get to the final answer efficiently.

Power Rule

The Power Rule is perhaps one of the most important rules in differentiation. It states that for any function of the form \( x^n \), the derivative is \( nx^{n-1} \).

Let's apply this to our function:

• For the term \( s^3 \), the derivative is \( 3s^2 \).
  That's because \( 3 \) (the exponent) comes down and becomes a multiplier, and we reduce the exponent by one.
• For the term \( s^2 \), the derivative is \( 2s \). The same rule applies here as well, with the exponent \( 2 \) becoming a multiplier and the exponent reducing by one.

So, \( \frac{d}{ds}(s^3 - s^2) = 3s^2 - 2s \).

By applying the Power Rule, we make the process of differentiating each term simple and straightforward.