Course:MATH110/Archive/2010-2011/003/Notes/The derivative as a function

If $a$ is a point in the domain of a function $f$, then the derivative of the function at this point is given by the limit:

$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}$

If the limit doesn't exist, we say that the function is not differentiable at the point $a$, which means that there is no tangent line for some reasons (can you find some examples of this?).

Since we can compute the derivative of a function at any given point and get a number as a result, it makes sense to create a function $f^{\prime }$, which we'll simply call the derivative of $f$, giving us precisely these values.

Example
What is the derivative of the function $f(x)=x^4$? We simply compute the derivative at a generic point $x$. $f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(a+h}{}$