Problem 16 - Calculus for Biology and Medicine

Question

Use the product rule to find the derivative with respect to the independent variable. $$ h(s)=\left(4-3 s^{2}+4 s^{3}\right)^{2} $$

Step-by-Step Solution

Answer

The derivative is \( h'(s) = 192s^5 - 384s^4 + 168s^3 - 144s^2 \).

Step 1: Identify function structure

The function given is \( h(s) = (4 - 3s^2 + 4s^3)^2 \). We can recognize that this is a composite function, specifically a power function applied to a polynomial. To proceed with differentiation, we need to apply the chain rule first and then incorporate the product rule.

Step 2: Apply the Chain Rule

According to the chain rule, the derivative of \( h(s) \) with respect to \( s \) can be found using \( \frac{d}{ds} [f(g(s))] = f'(g(s)) \cdot g'(s) \). Here, \( f(u) = u^2 \) and \( u = 4 - 3s^2 + 4s^3 \). First, differentiate \( f(u) \): \( f'(u) = 2u \). Then substitute back to get \( f'(g(s)) \): \( 2(4 - 3s^2 + 4s^3) \).

Step 3: Differentiate Inner Function

Now, differentiate the inner function \( g(s) = 4 - 3s^2 + 4s^3 \) with respect to \( s \). The derivative is: \( g'(s) = 0 - 6s + 12s^2 = 12s^2 - 6s \).

Step 4: Combine Chain Rule Components

To find the derivative using the chain rule, multiply \( f'(g(s)) \) by \( g'(s) \): \( h'(s) = 2(4 - 3s^2 + 4s^3) \cdot (12s^2 - 6s) \).

Step 5: Simplify the Expression

Expand the expression: \( h'(s) = 2 \cdot (4 - 3s^2 + 4s^3) \cdot (12s^2 - 6s) \). This will require distributing terms: \( 2((4) \cdot (12s^2) - (4) \cdot (6s) - (3s^2) \cdot (12s^2) + (3s^2) \cdot (6s) + (4s^3) \cdot (12s^2) - (4s^3) \cdot (6s)) \).

Step 6: Calculate Each Product

Calculate each component: \( -72s^2 + 48s^3 - 144s^4 + 36s^3 + 96s^5 - 48s^4 \).

Step 7: Combine Like Terms

Combine the terms: \( 96s^5 - 144s^4 - 48s^4 + 48s^3 + 36s^3 - 72s^2 \). This simplifies to \( 96s^5 - 192s^4 + 84s^3 - 72s^2 \).

Step 8: Include All Coefficients

Factor in the \( 2 \) multiplier from Step 4: \( h'(s) = 2 \times (96s^5 - 192s^4 + 84s^3 - 72s^2) \). Expand to get the final derivative: \( h'(s) = 192s^5 - 384s^4 + 168s^3 - 144s^2 \).

Key Concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus used to differentiate composite functions, which are functions within functions. In the original exercise, we dealt with a composite function of the form \( h(s) = (4 - 3s^2 + 4s^3)^2 \). This required us to apply the Chain Rule to correctly differentiate.To apply the Chain Rule, follow these steps:

• Identify the outer and inner functions. Here, the outer function \( f(u) = u^2 \) and the inner function \( g(s) = 4 - 3s^2 + 4s^3 \).
• Find the derivative of the outer function, \( f'(u) = 2u \), and then replace \( u \) with \( g(s) \) to get \( f'(g(s)) = 2(4 - 3s^2 + 4s^3) \).
• Next, differentiate the inner function \( g(s) \) to get \( g'(s) = 12s^2 - 6s \).
• Finally, multiply these derivatives: \( h'(s) = f'(g(s)) \cdot g'(s) = 2(4 - 3s^2 + 4s^3) \cdot (12s^2 - 6s) \).

Breaking down a composite function like this ensures you can tackle complex derivatives with simplicity.

Power Function

A Power Function is any expression of the form \( x^n \), where \( n \) is a constant. In differentiation, the Power Rule simplifies finding derivatives of such functions. It states that if you have \( f(x) = x^n \), then the derivative \( f'(x) = nx^{n-1} \).Applying the Power Rule to functions simplifies using calculus, especially when combined with other rules like the Chain Rule. In the given problem, the outer function \((4 - 3s^2 + 4s^3)^2\) is a power function with \( n = 2 \). Recognizing this helps leverage the Chain Rule, as the outer function’s derivative is quite straightforward at \( 2u \).Thus, whenever you encounter a power of a function, consider simplifying your problem with the Power Rule first. It makes the differentiation process faster and more efficient.

Polynomial Differentiation

Polynomial differentiation involves finding the derivative of a polynomial function, which is a sum of power functions. In the context of the original exercise, the expression \( g(s) = 4 - 3s^2 + 4s^3 \) is a polynomial.To differentiate a polynomial function:

• Use the Power Rule for each term separately: for example, \( -3s^2 \) becomes \( -6s \), and \( 4s^3 \) becomes \( 12s^2 \).
• Combine the results to get the derivative of the complete polynomial: \( g'(s) = 0 - 6s + 12s^2 = 12s^2 - 6s \).

Polynomial differentiation allows for straightforward application because each term is differentiated independently and then recombined. Understanding this process is crucial for tackling complex derivatives where polynomials appear, like in the exercise provided. This makes solving such calculus problems more manageable by breaking them down into smaller, more approachable parts.