Problem 2 - Calculus for Biology and Medicine

Question

Use the product rule to find the derivative with respect to the independent variable. $$ f(x)=\left(2 x^{3}-1\right)\left(3+2 x^{2}\right) $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 20x^4 + 18x^2 - 4x \).

Step 1: Identify the functions

The function given is the product of two functions: \( u(x) = 2x^3 - 1 \) and \( v(x) = 3 + 2x^2 \). We need to find the derivatives of these two functions individually before applying the product rule.

Step 2: Find the derivative of the first function

Calculate \( u'(x) \) by differentiating \( u(x) = 2x^3 - 1 \) with respect to \( x \). The derivative is \( u'(x) = 6x^2 \).

Step 3: Find the derivative of the second function

Calculate \( v'(x) \) by differentiating \( v(x) = 3 + 2x^2 \) with respect to \( x \). The derivative is \( v'(x) = 4x \).

Step 4: Apply the product rule

The product rule states that the derivative of a product of two functions, \( f(x) = u(x)v(x) \), is \( f'(x) = u'(x)v(x) + u(x)v'(x) \). Substitute the known values: \( f'(x) = (6x^2)(3 + 2x^2) + (2x^3 - 1)(4x) \).

Step 5: Simplify the expression

Expand each term: \( (6x^2)(3) + (6x^2)(2x^2) + (2x^3)(4x) - (1)(4x) \). This results in \( 18x^2 + 12x^4 + 8x^4 - 4x \).

Step 6: Combine like terms

Combine like terms in the expression: \( f'(x) = 12x^4 + 8x^4 + 18x^2 - 4x = 20x^4 + 18x^2 - 4x \). This is the final derivative.

Key Concepts

Derivative Calculation

Calculating derivatives is a fundamental skill in calculus. A derivative represents the rate at which a function changes as its input changes. In this problem, we need to find the derivative of a product of functions using the product rule.
To start, we have the function given by\[ f(x) = (2x^3 - 1)(3 + 2x^2) \].
We've identified it as a product of two separate functions - \( u(x) = 2x^3 - 1 \) and \( v(x) = 3 + 2x^2 \).

• We calculate the derivative of each of these functions separately.
  
• First, find \( u'(x) \) by differentiating \( u(x) \). The derivative turns out to be \( 6x^2 \).
  
• Next, find \( v'(x) \) by differentiating \( v(x) \), resulting in \( 4x \).
  It's crucial to remember: when working with product rule, derivative potentials must be tackled individually before bringing them together.

Function Differentiation

Differentiation involves finding the derivatives, which inform us about the slopes of functions at any given point.
When dealing with more complex functions, such as products, it’s essential to break them down into simpler parts, differentiate each one, and then apply rules that tie them together, such as the product rule.
This is particularly useful when two functions are multiplied together, as each function independently influences the overall derivative.
In our original function:

• \( u(x) = 2x^3 - 1 \), which is a simple polynomial function. Differentiating it gives \( u'(x) = 6x^2 \).
  
• \( v(x) = 3 + 2x^2 \) is also straightforward. Its derivative is \( v'(x) = 4x \).
  This method allows us to manage complexity by focusing on smaller components before piecing together the full derivative.

Simplification of Expressions

After finding the derivatives using the product rule, the next step is simplifying the expression to make it more understandable.
After applying the product rule, your expression might be intricate:
\[ f'(x) = (6x^2)(3 + 2x^2) + (2x^3 - 1)(4x) \].
The aim is to simplify the expression by expanding the products and combining like terms:

• First, expanding: \( (6x^2 \times 3) + (6x^2 \times 2x^2) + (2x^3 \times 4x) - (1 \times 4x) \)
• This results in: \( 18x^2 + 12x^4 + 8x^4 - 4x \)
• Then, combine like terms: \( 12x^4 + 8x^4 + 18x^2 - 4x \)

The final simplified derivative is then \
\[ f'(x) = 20x^4 + 18x^2 - 4x \].
Simplifying expressions allows us to present the derivative in the cleanest form, which is easier for further analysis or graphing.