Differentiation: The derivative
The tangent line
Consider the graph of the function #f(x)=2\sqrt{x}# pictured to the right. We can draw a straight line #\orange{k}# for the difference quotient at #\blue{a}# with difference #\green{h}# which moves through the two points #\rv{\blue{a},f(\blue{a})}# and #[\blue{a}+\green{h},f(\blue{a}+\green{h})]#, like we did with linear functions.
If we take a smaller #\green{h}#, we see that line #\orange{k}# around point #\blue{a}# is increasingly similar to the graph.
The tangent line is an important concept. This is a straight line through a point on a graph that has the same slope as the graph at that point.
The tangent line
We call a line that touches a graph at point #\blue{P}# the tangent line at that point #\blue{P}#. This means that the slope of the tangent line and of the graph at point #\blue{P}# are equal, making them look very similar around point #\blue{P}#.
The tangent line #\orange{k}# is a linear function #\orange{k(x)}=\orange{ax+b}#. Here the slope #a# is equal to the slope of the graph at point #\blue{P}#.
In the image to the right, you can move the point #\blue{P}# yourself to see what the tangent line #\orange{k}# to the graph #f(x)=\tfrac{1}{5}x^3-2x# is at point #\blue{P}#.
Which of these lines is/are tangent lines of the graph #f(x)#?
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