How do you find the slope of the tangent line for #f(x) = 3x^2# at (1,3)?
The slope is
I will assume that you have not yet been taught the rules (shortcuts) for finding derivatives. So, we will use a definition.
The slope of the line tangent to the graph of the function
(Each author,teacher,presenter needs to choose one definition as the 'official' definition. Many will immediately mention other possibilities as 'equivalents'.)
For this question we have
(Note that substitution gets us the indeterminate form
# = lim_(xrarr1) (3(x^2-1))/(x-1)#
# = lim_(xrarr1) (3(x+1)(x-1))/(x-1)#
The expression whose limit we want is equal to
#lim_(xrarr1) (3(x+1)(x-1))/(x-1) = lim_(xrarr1) 3(x+1) = 6#
The slope of the tangent we were asked about is
The slope of the tangent at