91Ó°ÊÓ

The chain rule is a basic but powerful concept in calculus that helps us to find the derivative of composite functions. Suppose you have a function composed of two functions: \( g(h(x)) \). The chain rule states that the derivative of this composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function:

• Derivative \( g'(h(x)) \) at \( h(x) \)
• Times the derivative of \( h(x) \), which is \( h'(x) \)

This can be written as \( (g(h(x)))' = g'(h(x)) \cdot h'(x) \). In simpler terms, you "chain" or link the derivatives of the functions.
Remember, whenever you see functions nested within each other, think of the chain rule. It helps break down complex derivative problems into manageable parts.

The change of base formula is a handy mathematical tool that allows you to rewrite logarithms in different bases. This is essential since most calculators and many mathematical systems support only base 10 (common logarithms) and base \( e \) (natural logarithms). The formula is:

• To change \( \log_a b \) to base \( c \), use: \( \log_a b = \frac{\log_c b}{\log_c a} \)

In the given problem, you had \( \log_2 x \). By using the change of base formula with natural logarithms (base \( e \)), it becomes \( \frac{\ln x}{\ln 2} \).
This transformation allows us to employ known derivative rules for natural logarithms to find derivatives easily. By changing the base, we turn a less common logarithm into a more common form, which can be easier to manipulate in calculus problems.

The quotient rule is a method used in calculus to find the derivative of a quotient of two functions. When you have a function in the form \( \frac{u(x)}{v(x)} \), the derivative \( f'(x) \) can be determined using:

• The formula: \( \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} \)

This rule is particularly useful when logarithmic functions are involved, as they often appear as quotients after applying the change of base formula.
For instance, in the problem, \( u(x) = \ln x \) and \( v(x) = \ln 2 \) were applied in this way. Since \( \ln 2 \) is a constant, its derivative \( v'(x) \) is zero, simplifying the calculation.
Thus, the quotient rule is a crucial tool in differentiation, simplifying complex fractional derivatives into straightforward steps by evaluating the individual components involved.