Example 1
Consider "f"("x") = 5:
:
The derivative of a constant function is zero.
Example 2
Consider the graph of . If the reader has an understanding of algebra and the Cartesian coordinate system, the reader should be able to independently determine that this line has a slope of 2 at every point. Using the above quotient (along with an understanding of the limit, secant, and tangent) one can determine the slope at (4,5):
:
The derivative and slope are equivalent.
Example 3
Via differentiation, one can find the slope of a curve. Consider :
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For any point "x", the slope of the function is .
Example 4
Consider :
:
Example 5
The same as the previous example, but now we search the derivative of the derivative.
Consider :
: