Problem 17 - Calculus and its applications

Question

Differentiate each function $$ f(x)=\left(3+x^{3}\right)^{5}-\left(1+x^{7}\right)^{4} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 15x^2(3+x^3)^4 - 28x^6(1+x^7)^3 \).

Step 1: Identify the Outer Functions

Look at the given function, which is \[ f(x)=\left(3+x^{3}\right)^{5}-\left(1+x^{7}\right)^{4}. \]This expression consists of two terms, \[ \left(3+x^{3}\right)^{5} \quad \text{and} \quad \left(1+x^{7}\right)^{4}. \]Both are composition of functions, requiring the use of the chain rule to differentiate.

Step 2: Apply the Chain Rule to the First Term

Differentiate the first term \( \left(3+x^{3}\right)^{5} \) using the chain rule.The chain rule states that if \( y = g(h(x)) \), then \( \frac{dy}{dx} = g'(h(x)) \cdot h'(x) \).Set \( g(u) = u^5 \) and \( h(x) = 3+x^3 \).The derivative \( g'(u) \) is \( 5u^4 \), and \( h'(x) = 3x^2 \).So, the derivative of the first term is: \[ 5(3+x^3)^4 \cdot 3x^2. \]

Step 3: Apply the Chain Rule to the Second Term

Now, differentiate the second term \( \left(1+x^{7}\right)^{4} \) using the chain rule.Set \( g(v) = v^4 \) and \( h(x) = 1+x^7 \).The derivative \( g'(v) \) is \( 4v^3 \), and \( h'(x) = 7x^6 \).Thus, the derivative of the second term is: \[ 4(1+x^7)^3 \cdot 7x^6. \]

Step 4: Combine the Derivatives of Both Terms

Combine the derivatives of both terms obtained from the chain rule. Thus, the complete derivative is:\[ f'(x) = 5(3+x^3)^4 \cdot 3x^2 - 4(1+x^7)^3 \cdot 7x^6. \]Simplifying this expression, you get:\[ f'(x) = 15x^2(3+x^3)^4 - 28x^6(1+x^7)^3. \]

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that is used to find the rate at which a function changes at any given point. It is essentially the process of finding a derivative. In simpler terms, if you have a function represented as a curve, differentiation helps you determine the slope of that curve at any point.

Imagine driving a car; the speedometer shows how fast you're going at any given instant. Similarly, the derivative of a function shows how fast the function's value is changing concerning its input, variable often denoted as \( x \).

• To differentiate a function involves applying specific rules, such as the power rule, product rule, quotient rule, and chain rule.
• The rules are designed to simplify the process based on the form and complexity of the function.
• In our given exercise, we primarily use the chain rule for differentiation due to the composition of functions within the expression.

Composition of Functions

Composition of functions is like layering functions one inside the other. When dealing with complex functions, often a function is built by applying one function to the result of another.

For example, in your daily routine, consider how you wake up in the morning (function 1), then brush your teeth (function 2). Each step affects the other, creating a sequence—a composition.

In mathematics, if you have two functions \( g(x) \) and \( h(x) \), composing them results in a new function denoted as \( g(h(x)) \).

• This allows complex functions to be broken down into simpler, manageable parts.
• In the exercise expression \( (3+x^3)^5 - (1+x^7)^4 \), we observe two compositions:\( \left(3+x^3\right)^5 \) and \( \left(1+x^7\right)^4 \).
• By identifying outer and inner functions, we use differentiation rules, like the chain rule, to work on each part separately before combining results.

Derivative of a Polynomial

Polynomials are algebraic expressions that include terms with variables raised to whole number exponents. Derivatives of polynomials can be efficiently found using straightforward rules.

The power rule is particularly useful: if \( h(x) = x^n \), then the derivative \( h'(x) = nx^{n-1} \).

The expression given, \( (3 + x^3)^5 \) and \( (1 + x^7)^4 \), involves taking the derivative of composite functions. Here, each component is a simple polynomial, easily differentiated using the power rule.

• We break down each polynomial term, like \( 3x^2 \) derived from \( x^3 \) and \( 7x^6 \) from \( x^7 \), using the power rule.
• Applying the chain rule provides a structured way to manage the polynomial within another function, making finding derivatives cleaner.
• This approach keeps polynomial derivatives accessible, even when composing them within more complex expressions.