Problem 33 - Applied Calculus

Question

33-38. Find the derivative of each function. $$ \left(x^{5}-3 x^{3}+x-1\right)^{4} $$

Step-by-Step Solution

Answer

The derivative is \( 4 (x^5 - 3x^3 + x - 1)^3 (5x^4 - 9x^2 + 1) \).

Step 1: Identify the Outer and Inner Functions

The given function is \( \left( x^5 - 3x^3 + x - 1 \right)^4 \). Notice that this is a composition of two functions: the outer function \( u^4 \) where \( u = x^5 - 3x^3 + x - 1 \), and the inner function \( x^5 - 3x^3 + x - 1 \).

Step 2: Differentiate the Outer Function

Using the chain rule, differentiate the outer function \( u^4 \) with respect to \( u \). The derivative is \( 4u^3 \).

Step 3: Differentiate the Inner Function

Now, differentiate the inner function \( x^5 - 3x^3 + x - 1 \) with respect to \( x \). The derivative is \( 5x^4 - 9x^2 + 1 \).

Step 4: Apply the Chain Rule

According to the chain rule, the derivative of the composition is the product of the derivatives from steps 2 and 3. Multiply the derivative of the outer function (\( 4u^3 \)) by the derivative of the inner function (\( 5x^4 - 9x^2 + 1 \)). Thus, the derivative of the whole function is \( 4 \left( x^5 - 3x^3 + x - 1 \right)^3 \cdot \left( 5x^4 - 9x^2 + 1 \right) \).

Key Concepts

Chain Rule

The Chain Rule in calculus is an essential tool for differentiating composite functions. A composite function is one that involves one function nested inside another. In our exercise, we faced a function of the form \( (x^5 - 3x^3 + x - 1)^4 \), which clearly has an outer function \( u^4 \) where the inner function is \( u = x^5 - 3x^3 + x - 1 \). The beauty of the chain rule lies in its simplicity to systematically handle these complexities.

Here's an easy way to think about the chain rule: when differentiating \( y = f(g(x)) \), we differentiate the outer function \( f \) with respect to \( u \) and the inner function \( g \) with respect to \( x \), then multiply these derivatives together.

• Differentiating the outer function \( u^n \) gives us \( nu^{n-1} \).
• The derivative of the composite function \( f(g(x)) \) is then \( f'(g(x)) \cdot g'(x) \).

The solution to our problem involves this direct application of the chain rule, enabling us to manage and work through complicated nested functions effectively.

Derivative of Polynomials

Differentiating polynomial functions can sometimes seem daunting due to the terms involved, but it follows a structured approach. Polynomial differentiation relies on a straightforward rule: for a term \( x^n \), the derivative is \( nx^{n-1} \). This rule derives from the power rule in calculus, which provides an easy-to-use formula for finding the rate of change of polynomials.

When we look at the inner function in the exercise, \( x^5 - 3x^3 + x - 1 \), each term is handled independently according to the power rule:

• The derivative of \( x^5 \) is \( 5x^4 \).
• The derivative of \( -3x^3 \) is \( -9x^2 \).
• The derivative of \( x \) is \( 1 \).
• A constant \( -1 \) drops out as its derivative is \( 0 \).

By applying these rules across each term, we simplify complex polynomials into manageable expressions, ensuring clarity and accuracy in calculations.

Calculus Problem Solving

Solving calculus problems involves understanding, applying, and linking different calculus concepts. Key to effective problem-solving is recognizing the type of problem, specifically which rules and formulas apply. The comprehensive approach to cracking such problems is by breaking down the problem into steps, as demonstrated through the chain rule and polynomial differentiation in our exercise.

Let's walk through how we tackled this:

• First, identify the structure of the function. Is it a composite function? What are the inner and outer functions?
• Second, determine the appropriate rules: here, it's crucial to invoke the chain rule for the composition and apply the power rule for polynomials.
• Finally, methodically perform calculations step by step. Take each derivative as per the rule, then combine them as required by the specific calculus principles.

This systematic approach of dissecting a problem into smaller, foremost understandable parts, and the application of the correct calculus rules ensures you arrive at the solution accurately and efficiently. Regular practice with these steps strengthens understanding and builds confidence in tackling a variety of calculus problems.