91Ó°ÊÓ

When working with composite functions, we aim to find the derivative of a function that is made by combining two or more simpler functions. In this context, we have two functions: \( k(t) = 4.3t^3 - 2t^2 + 4t - 12 \) and \( t(x) = \ln x \). The composite function is expressed as \( k(t(x)) \), which means we're plugging \( \ln x \) into the function \( k(t) \) wherever we see \( t \). After substituting, the resulting expression is \( k(t(x)) = 4.3(\ln x)^3 - 2(\ln x)^2 + 4(\ln x) - 12 \). To find the derivative of this composite function with respect to \( x \), we differentiate every term individually. This requires an understanding of derivatives of polynomials as well as the natural logarithm function. The key takeaway is that when dealing with composite functions, especially when finding derivatives, substitution and careful differentiation are crucial for obtaining the correct derivative.

The Chain Rule is an essential concept when dealing with composite functions. It provides us the means to differentiate a function composed of other functions. For a function \( k(t(x)) \), the Chain Rule can be understood as:

• First, differentiate the outer function \( k(t) \) while treating \( t(x) \) as an inner function.
• Then, multiply by the derivative of the inner function \( t(x) \). This is represented as \( \frac{d}{dx}[t(x)] \).

In our exercise, each term in \( k(t(x)) = 4.3(\ln x)^3 - 2(\ln x)^2 + 4(\ln x) - 12 \) is differentiated using this rule. For example, differentiating \( 4.3(\ln x)^3 \) involves applying the power rule and then multiplying by the derivative of \( \ln x \), which is \( \frac{1}{x} \). This chain of multiplication reinforces the "chain" process that the rule is named after. By using the Chain Rule accurately, we ensure that our differentiated composite function respects the influence of each "piece" of the function as they change with respect to \( x \).

Function Substitution is a technique where we replace a variable within a function with another function. This is a key step in forming and working with composite functions, as seen in the provided exercise. Here, the substitution of \( t(x) = \ln x \) into \( k(t) = 4.3t^3 - 2t^2 + 4t - 12 \) involves replacing every instance of \( t \) with \( \ln x \). This results in forming the composite function \( k(t(x)) = 4.3(\ln x)^3 - 2(\ln x)^2 + 4(\ln x) - 12 \) where each term now involves \( \ln x \) instead of \( t \). Each substitution step transforms our approach to differentiation. Since the function now involves \( \ln x \), the substitution changes how we apply derivation rules, such as polynomial rules and logarithmic rules. Understanding Function Substitution allows us to connect different mathematical functions seamlessly and lays the foundation for further calculus operations like differentiation and integration. It's an empowering technique that simplifies complex expressions into tractable forms.