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A function $ f$ is differentiable at the point $ a$ if the limit

$\displaystyle f^\prime(a)= \lim_{h\to0} \frac{f(a+h)-f(a)}{h} $

exists. This limit is called the derivative of $ f$ at $ a$.


Geometrically, differentiability means that the slopes of the secants converge to the slope of the tangent given by

$\displaystyle y=f(a)+f^\prime(a)(x-a)

We also write

$\displaystyle f^{\prime}(x)=\frac{d}{dx}f(x)=\frac{dy}{dx}

with $ y=f(x)$. This notation symbolizes the limit $ \Delta x\to0$ for the difference quotient.

Higher derivatives are denoted by $ f^{\prime\prime},f^{\prime\prime\prime},\ldots$ or $ f^{(2)},f^{(3)},\ldots$, respectively.

We say that a function $ f$ is differentiable on a set $ D$ if $ f^\prime(x)$ exists for all $ x\in D$.

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  automatically generated 6/14/2016