Part 1: The derivative at a specific point
Use the definition of the derviative to compute the derivative of f(x)=1−4x2
at the specific point x=2
. Evaluate the limit by using algebra to simplify the difference quotient (in first answer box) and then evaluating the limit (in the second answer box).
f′(2)=limh→0(f(2+h)−f(2)h)=limh→0(

)=

.
Part 2: The derivative function
Use the definition of the derivative to compute the derivative of the function f(x)=1−4x2
at an arbitrary point x
. Evaluate the limit by using algebra to simplify the difference quotient (in first answer box) and then evaluating the limit (in the second answer box).
f′(x)=limh→0(f(x+h)−f(x)h)=limh→0(

)=

.
Part 3: The tangent line
Now let’s calculate the tangent line to the function f(x)=1−4x2
at x=4
.
By using f′(x)
from part 2, the slope of the tangent line to f
at x=4
is f′(4)=

.
The tangent line to f
at x=4
passes through the point (4,f(4))=

on the graph of f
. (Enter a point in the form (2, 3) including the parentheses.)
An equation for the tangent line to f
at x=4
is y=

.