Part 2/5:
- The derivative of ( f'(u) ) can be calculated using the power rule:
[ f'(u) = 100u^{99} ]
- The derivative of ( u ) with respect to ( x ) is:
[ u' = 2x ]
Establishing the Derivative Definition
To move towards proving the chain rule, we must utilize the definition of a derivative. The derivative ( f'(x) ) is defined as:
[ f'(x) = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} ]
where ( \Delta y = f(x + \Delta x) - f(x) ). This limit mechanic reflects how instantaneous rate of change can be understood as the slope of the function over an infinitesimal interval.
In our case, ( \Delta y ) can be expressed as:
[ \Delta y = f(x + \Delta x) - f(x) ]
with a corresponding increment defined as:
[ \epsilon = \frac{\Delta y}{\Delta x} - f'(x) ]