newhite1 2 years ago In Problem Set 1: Problem 1J-2: How does d/dx cos(x) when x is PI/2 = Cos(x)-Cos(PI/2)/(x-PI/2). Shouldn't the numerator be Cos(x-PI/2)-Cos(PI/2)? (Difference Quotient)

1. NoelGreco

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2. newhite1
We define derivative this way: $f'(x) = \lim_{\Delta x \rightarrow 0 } \frac { f(x + \Delta x) - f(x) } { \Delta x }$ Substitute $\Delta x = x_0 - x$ to get $f'(x) = \lim_{ x \rightarrow x_0 } \frac { f(x + (x_0 - x)) - f(x) } { x_0 - x } \\ = \lim_{ x \rightarrow x_0 } \frac { f(x_0) - f(x) } { x_0 - x } \\ = \lim_{ x \rightarrow x_0 } \frac { - (f(x) - f(x_0)) } { -(x - x_0) } \\ = \lim_{ x \rightarrow x_0 } \frac { f(x) - f(x_0) } { x - x_0 }$ Therefore $\cos'(x) = \lim_{x \rightarrow \pi/2} \frac { \cos ( x ) - \cos (\pi /2) } { x - \pi/2} = \lim_{x \rightarrow \pi/2} \frac { \cos ( x ) - \cos( \pi /2) } { x - \pi/2}$