Part 2/5:
To understand how to differentiate logarithmic functions, it is crucial to grasp what logarithms represent. A logarithm ( \log_a(x) ) answers the question: "To what exponent must the base ( a ) be raised to produce ( x )?" This can be expressed as ( a^y = x ), which rearranges to give ( y = \log_a(x) ).
Using implicit differentiation, we can find the derivative of ( y ) with respect to ( x ). By applying the rules of implicit differentiation to the equation ( a^y = x ), we can derive a formula for ( \frac{dy}{dx} ).
Derivative of ( \log_a(x) )
Let's start deriving the derivative of ( \log_a(x) ).
Starting with the equation ( a^y = x ), we take the derivative with respect to ( x ):
[
\frac{d}{dx}(a^y) = \frac{d}{dx}(x)
]
Using the chain rule, we acquire: