Example1:
Find the slopeof the tangentline to the curvef(x) =x^3 passingthroughpointP(2, 8).
Solution:
SinceP(x 0 ,y 0 ) = (2, 8), usingthe slopeof the tangentequation,
we get
Thusthe slopeof the tangentline is 12. Usingthe point-slopeformulaabove,
or
Next we are interestedin findinga formulafor the slopeof the tangentline atany pointon the curvef. Such
a formulawouldbe the sameformulathat we are usingexceptwe replacethe constantx 0 by the variable
x. This yields
We denotethis formulaby
where is read “ primeof .” The next exampleillustrateits usefulness.
Example2:
If find and use the resultto find the slopeof the tangentline at and
Solution: