Part 2/6:
This product of derivatives highlights how the outer function’s rate of change depends on the inner function. This technique is vital for understanding more complex relationships between functions.
Example 1: The Derivative of a Nested Function
Let's begin with an intricate nested function:
( f(x) = \sin(\cos(\tan(x))) )
To find the derivative, we proceed by applying the Chain Rule step by step.
The outer function is ( \sin(u) ); thus, its derivative is ( \cos(u) ) where ( u = \cos(\tan(x)) ).
Next, we move to the middle function ( \cos(v) ), and its derivative is ( -\sin(v) ) where ( v = \tan(x) ).
Finally, the innermost function ( \tan(x) ) has a derivative of ( \sec^2(x) ).
Putting this all together, we have:
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