Problem 18 - Calculus and its applications

Question

Differentiate each function $$ f(x)=\left(1+x^{3}\right)^{3}-\left(2+x^{8}\right)^{4} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 9x^2(1+x^3)^2 - 32x^7(2+x^8)^3. \)

Step 1: Apply the Power Rule and Chain Rule

We need to differentiate the function \[ f(x) = \left(1+x^{3}\right)^{3}-\left(2+x^{8}\right)^{4}. \]This function is composed of two parts separated by a minus sign. We begin by differentiating each part separately using the chain rule. The chain rule states that\[ \frac{d}{dx} g(h(x)) = g'(h(x)) \cdot h'(x). \]

Step 2: Differentiate the First Part

Differentiate the first part \( \left(1+x^3\right)^3 \). Set \( u = 1 + x^3 \) so that the function becomes \( u^3 \). According to the chain rule, the derivative is:\[ \frac{d}{dx}(u^3) = 3u^2 \frac{d}{dx}(1 + x^3). \]The derivative of \( 1 + x^3 \) is \( 3x^2 \). Thus, the derivative of the first part is\[ 3(1+x^3)^2 \cdot 3x^2 = 9x^2(1+x^3)^2. \]

Step 3: Differentiate the Second Part

Now, differentiate the second part \( \left(2+x^8\right)^4 \). Set \( v = 2 + x^8 \) so the function becomes \( v^4 \). Applying the chain rule, we have:\[ \frac{d}{dx}(v^4) = 4v^3 \frac{d}{dx}(2 + x^8). \]The derivative of \( 2 + x^8 \) is \( 8x^7 \). Hence, the derivative of the second part is\[ 4(2+x^8)^3 \cdot 8x^7 = 32x^7(2+x^8)^3. \]

Step 4: Combine the Derivatives

Combine the derivatives of both parts to find the derivative of the entire function:\[ f'(x) = 9x^2(1+x^3)^2 - 32x^7(2+x^8)^3. \]

Key Concepts

Power Rule

The power rule is a fundamental rule in calculus used for finding derivatives. It states that if you have a function of the form \[ f(x) = x^n, \]then the derivative \( f'(x) \) is given by \[ f'(x) = n \cdot x^{n-1}. \]This rule simplifies the process of differentiating polynomial terms by bringing the exponent down as a coefficient and reducing the exponent by one.

For example, if you need to differentiate \( x^3 \), applying the power rule gives - The derivative of \( x^3 \) is \( 3x^2 \).
Additionally, the power rule can be used within larger functions such as polynomial expressions or when using the chain rule for nested functions. Understanding the power rule can make calculating derivatives of more complex expressions much more manageable.

Chain Rule

The chain rule is a powerful technique in calculus used for differentiating composite functions. When a function is inside another function, the chain rule comes into play. It states that if you have a composite function \[ f(g(x)), \]the derivative is \[ f'(g(x)) \cdot g'(x). \]
This means you differentiate the outer function with respect to the inner function first, and then multiply by the derivative of the inner function. It's like peeling an onion, layer by layer, until you get to the core.

For instance, given a function \( (1 + x^3)^3 \), set \( u = 1 + x^3 \) to find the derivative using the chain rule. This turns the function into \( u^3 \). First, find the derivative \( 3u^2 \), then multiply by the derivative of \( u \), which is \( 3x^2 \). Thus, the result is \( 9x^2(1+x^3)^2 \).

Using the chain rule becomes crucial when dealing with nested functions like those in our original exercise, allowing us to handle intricate expressions efficiently.

Derivative of Polynomial Functions

Polynomial functions are expressions composed of variables and coefficients, featuring terms like \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \). Differentiating polynomial functions relies on applying the power rule to each term separately. Since polynomials are just sums and differences of terms of the form \( x^n \), the power rule can be used straightforwardly.

Let's take the term \( x^8 \) as an example:

• Applying the power rule, the derivative \( \frac{d}{dx}x^8 = 8x^7 \).
• Each term is differentiated independently, and the results are combined.

When dealing with more complex expressions, like in our exercise, combinations of polynomial functions often require additional rules. For instance, the chain rule was necessary since sub-expressions within brackets required differentiating.

Understanding how to differentiate polynomial functions is essential because these functions model many real-world situations, making their derivatives practical for predicting changes and behavior of systems.