# Derivatives

Function:

x = 1

Show Tangent Line at x
First Derivative
Complete
Second Derivative
Complete

The first derivative $$f'(x)$$ of a function $$f(x)$$ is defined as the limit $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ At a particular value $$a$$, we can also define $$f'(a)$$ as $f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$ if this limit exists. This limit will be equal to the slope of the tangent line to $$f(x)$$ at $$x=a$$.
The second derivative $$f''(x)$$ is found by taking the derivative of the first derivative. That is, the value of $$f''(x)$$ is the slope of the tangent line to the graph of $$f'(x)$$ at $$x$$.
Choose a function to work with, and use the slider to adjust the current value of $$x$$. The tangent line to $$f(x)$$ at $$x$$ will be displayed, along with the graphs of $$f'(x)$$ and $$f''(x)$$ if these options are enabled.