Problem 3 - Calculus and its applications

Question

Differentiate each function $$ y=(7-x)^{55} $$

Step-by-Step Solution

Answer

The derivative is \( y' = -55(7-x)^{54} \).

Step 1: Identify the function's form

The function given is \( y = (7-x)^{55} \). This is a power function where a binomial expression \((7-x)\) is raised to the power of 55. This form suggests the use of both the general power rule for differentiation and the chain rule.

Step 2: Apply the chain rule

The chain rule states that if you have a composite function \( y = (u(x))^n \), then \( \frac{dy}{dx} = n \cdot (u(x))^{n-1} \cdot \frac{du}{dx} \). In our problem, \( u(x) = 7-x \), and \( n = 55 \). First, find the derivative of the inner function \( u(x) = 7-x \), which is \( \frac{du}{dx} = -1 \).

Step 3: Differentiate using the power rule

Use the power rule for differentiation: \( \frac{d}{du}(u^n) = n \cdot u^{n-1} \). Apply this to the outer function: \( y = (7-x)^{55} \). Thus, \( \frac{dy}{dx} = 55 \cdot (7-x)^{54} \).

Step 4: Multiply by the derivative of the inner function

According to the chain rule, multiply the derivative of the outer function by the derivative of the inner function. Thus, \( \frac{dy}{dx} = 55 \cdot (7-x)^{54} \cdot (-1) \).

Step 5: Simplify the final expression

Simplify the expression for the derivative by multiplying through with \(-1\): \( \frac{dy}{dx} = -55 \cdot (7-x)^{54} \). This is the final, simplified derivative of the function.

Key Concepts

Understanding the Chain Rule

The chain rule is an essential technique in differentiation, especially when dealing with composite functions. At its core, the chain rule helps us differentiate functions that are nested within one another—also known as composite functions.
This is common when dealing with functions of the form \( f(g(x)) \). Here, the idea is to differentiate the outer function first and then multiply by the derivative of the inner function.
This multi-step process is crucial when our function isn't just a straightforward single-variable function.

For example, in our problem, we have the function \( y = (7-x)^{55} \). This is a composite function where \( u(x) = 7-x \) acts as the inner function, and \( (u(x))^{55} \) is the outer function. First, we differentiate the outer function using the power rule (which we will discuss next) and then find the derivative of the inner function \( u(x) = 7-x \), which is \( -1 \).
Finally, to find the derivative of the entire function, we multiply these derivatives together.

Exploring the Power Rule

The power rule is a straightforward formula used when differentiating functions in the form of \( x^n \). It states that if \( y = x^n \), then \( \frac{dy}{dx} = n \cdot x^{n-1} \). This means we bring down the exponent as a coefficient and decrease the exponent by one.

This rule simplifies the process of differentiation for polynomial functions, making it quick and efficient to apply.

• Bring down the exponent as a coefficient.
• Subtract one from the original exponent to get the new exponent.

In the context of our function \( y = (7-x)^{55} \), we can apply the power rule to differentiate the outer part of the function. Here, we treat \( (7-x) \) as a single unit or variable \( u \), so the power rule gives us \( 55 \cdot (7-x)^{54} \).
This term represents the derivative of the outer function before we apply the chain rule to incorporate the derivative of the inner function.

Understanding Composite Functions

Composite functions are those where one function is applied to the result of another, such as \( f(g(x)) \). In practice, this means that we have functions nested inside of each other. They appear often in real-world situations and calculus problems because many processes can be thought of as building blocks layered over each other.

For instance, consider the task of finding how various factors affect the final price of a product. Each factor affects another, creating a composite of effects rather than a single influence at work.

• Recognize that composite functions have an inner and outer function.
• Use the chain rule for differentiation as these functions are typically not straightforward.

In our specific exercise \( y = (7-x)^{55} \), \( (7-x) \) is the inner function \( u(x) \), and the whole expression is the outer function. Understanding which parts of the function relate to each other is crucial in applying the proper differentiation techniques effectively.