I am reading the following paper:
ergodic theory of chaos and strange attractors, by J.-P. Eckmann (can be easily downloaded)
My question is around equation (1.7), p. 619:
The separation of two initial points $x(0)$ and $x(0)'$ after time $N$ is
$$x(N) - x(N)' = f^N(x(0))-f^N(x(0)') \approx \bigg[ \frac{d}{dx}(f^N)(x(0)) \bigg] [x(0)-x(0)']$$
From this approximation, I get $$ f^N(x(0)) \approx \bigg[ \frac{d}{dx}(f^N)x(0) \bigg] x(0)$$ Consider $N = 1$,$$x(1) = f(x(0)) \approx \bigg[ \frac{d}{dx}(f)(x(0)) \bigg] x(0)$$
I am confused about what I remember in the following. Suppose $y = f(x)$ and I know $y_0 = f(x_0)$ so I can obtain the following $$y_0+\delta y = \bigg[\frac{d}{dx}(f)(x_0)\bigg]\delta x + f(x_0),$$ where $\frac{d}{dx}(f)(x_0)$ is the slope of $f$ at $x_0$.
What I should have is $\delta x$ instead of $x(0)$