# What’s the derivative of these equations in Taylor series?

I am struggling understand the linear approximation and Taylor.series. Could you give me a hint what are the derivatives of these functions?

$$a_2(x_1-x_0)^2 + a_3(x_1-x_0)^3?$$ If it’s stated that $$x_1=x_0$$.

• what have you tried that you couldn't find the derivatives? – pointguard0 Jul 5 at 8:35
• I mean, I can find a he derivatives, but I can’t get the whole intuition behind the Taylor Series concept. – Maria Lavrovskaya Jul 5 at 8:51
• Derivatives with respect to which variable? Anyway, for intuition about Taylor series, you should watch 3Blue1Brown's video on that topic: youtube.com/watch?v=3d6DsjIBzJ4 – Hans Lundmark Jul 5 at 9:34

Peano's theorem - Let $$f : (a,\,b) \to \mathbb{R}$$ and $$x_0 \in (a,\,b)$$. If $$f$$ is $$n$$ times derivable at $$x_0$$, the Taylor polynomial of order $$n$$ and centered at $$x_0$$, $$T_n(x) := \sum_{k = 0}^n \frac{f^{(k)}(x_0)}{k!}(x-x_0)^k$$ is the only polynomial of degree $$\le n$$ such that $$f(x) = T_n(x) + o\left((x-x_0)^n\right) \quad \text{for} \; x \to x_0$$ and also such that $$f^{(k)}(x_0) = T_n^{(k)}(x_0) \quad \text{for} \; k = 0,\,1,\,\dots,\,n.$$
Application example - Let $$f : (-3,\,3) \to \mathbb{R}$$ of law $$f(x) := e^x$$ and $$x_0 = 0$$, then it follows that:
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