Problem 36 - Calculus and its applications

Question

Differentiate each function $$ f(x)=\left(7 x^{4}+6 x^{3}-x\right)^{204} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 204(7x^4 + 6x^3 - x)^{203}(28x^3 + 18x^2 - 1) \).

Step 1: Understand the Problem

You need to differentiate the function \( f(x) = (7x^4 + 6x^3 - x)^{204} \). This should be done by applying the chain rule because the function is a composite function.

Step 2: Apply the Chain Rule

The chain rule states that if you have a composite function \( f(g(x)) \), then its derivative is \( f'(g(x)) \cdot g'(x) \). Here, \( g(x) = 7x^4 + 6x^3 - x \) and \( f(t) = t^{204} \), where \( t = g(x) \). The derivative of \( f(t) \) with respect to \( t \) is \( 204t^{203} \).

Step 3: Differentiate the Inside Function

Now find the derivative of the inside function \( g(x) = 7x^4 + 6x^3 - x \). The derivative is calculated term by term: \( g'(x) = 28x^3 + 18x^2 - 1 \).

Step 4: Combine the Derivatives

Apply the chain rule: \( f'(x) = f'(g(x)) \cdot g'(x) = 204(7x^4 + 6x^3 - x)^{203} \cdot (28x^3 + 18x^2 - 1) \).

Step 5: Final Expression

The final expression of the derivative is \( f'(x) = 204(7x^4 + 6x^3 - x)^{203}(28x^3 + 18x^2 - 1) \).

Key Concepts

Chain Rule

The chain rule is a powerful tool in calculus used when differentiating composite functions. Imagine you have a function nested within another function, like a matryoshka doll. This situation calls for the chain rule.

• When you have a composite function \( f(g(x)) \), consider \( f \) as the outer function and \( g \) as the inner function.
• The derivative of a composite function is found by multiplying the derivative of the outer function \( f \) evaluated at the inner function \( g(x) \), with the derivative of the inner function \( g(x) \).

So, the formula is \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \).
This rule is essential for handling functions where one function is applied to the results of another, allowing for the transformation from the inside out.

Polynomial Functions

Polynomial functions form one of the most basic types of mathematical expressions. They consist of variables, coefficients, and terms raised to non-negative integer powers. Often, the structure of a polynomial function is:

• General Format: \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)
• Each term includes a coefficient \( a \) and a power of \( x \).
• For example, \( 7x^4 + 6x^3 - x \) is a polynomial because it can be broken down into components like \( 7x^4 \), \( 6x^3 \), and \(-x \).

These functions are particularly simple to differentiate as they follow a straightforward pattern. Each term can be differentiated individually using power rule.

Derivative Calculation

Differentiation is the main process in calculus for finding the rate at which a function is changing at any point. For polynomial functions like \( g(x) = 7x^4 + 6x^3 - x \), differentiation is systematic and involves:

• Using the power rule, which is one of the most fundamental rules of differentiation. It states that the derivative of \( x^n \) is \( nx^{n-1} \).
• Applying the rule to each term in the polynomial: The derivative of \( 7x^4 \) is \( 28x^3 \), the derivative of \( 6x^3 \) is \( 18x^2 \), and for \(-x\), it is \(-1 \).
• Combining these to get \( g'(x) = 28x^3 + 18x^2 - 1 \).

In our composite function, the derivative of the outer function \( (g(x))^{204} \) is calculated separately using basic power rules, and then multiplied by the derivative of the inside function applying chain rule principles for accurate results.