91Ó°ÊÓ

The chain rule is a fundamental tool in calculus for differentiating composite functions. To understand it, consider a situation where one function is nested inside another. For example, if we have \( f(g(h(x))) \), the chain rule helps us find the derivative of this composition.

The chain rule states that if you have a composite function \( f(g(x)) \), the derivative is found by multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function. Mathematically, this can be expressed as:
\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]

In the given exercise, the function \(f(t) = \ln(t^2 + 7) \) is composed of the natural logarithm \( \ln(x) \) and an inner function \( t^2 + 7 \. \) To apply the chain rule, first differentiate the outer function, which is the natural logarithm, and then the inner function.

Logarithmic differentiation is a technique used to differentiate functions of the form \( \ln(u) \), where \( u \) is a function of \( x \. \) This method simplifies the differentiation process, especially when dealing with complicated products or quotients.

To apply logarithmic differentiation, use the fact that the derivative of \( \ln(u) \) with respect to \( t \) is given by:
\[ \frac{d}{dt} \ln(u) = \frac{1}{u} \cdot \frac{du}{dt} \]

In our exercise, the function \( f(t) = \ln(t^2 + 7) \) fits this form. We identify \( u = t^2 + 7 \) and use logarithmic differentiation. First, find the derivative of the outer function \( \ln(u) \), which is \( \frac{1}{u} \. \) Next, find the derivative of the inner function \( t^2 + 7 \), which is \( 2t \. \) Multiplying these results gives us the overall derivative.

Composite functions are functions made up of other functions. For example, if we have a function \( h(x) \) and another function \( g(x) \, \) a composite function would look like \( f(x) = h(g(x)) \. \) Differentiating composite functions requires the use of the chain rule, as mentioned earlier.

In the provided exercise, the function \( f(t) = \ln(t^2 + 7) \) is a composite function because it can be broken down into the composition of an outer function (the natural logarithm \( \ln \)) and an inner function \( (t^2 + 7) \. \) Here's a step-by-step breakdown for clarity:

• Identify the inner function: \( u = t^2 + 7 \)
• Identify the outer function: \( f(u) = \ln(u) \)
• Differentiate the outer function with respect to \( u \, \ (\frac{1}{u}) \)
• Differentiate the inner function with respect to \( t \, \ (2t) \)
• Combine the results: \( \frac{d}{dt}\ln(t^2 + 7) = \frac{1}{t^2 + 7} \cdot 2t = \frac{2t}{t^2 + 7} \)

Breaking down composite functions into their inner and outer parts makes using the chain rule straightforward.