Page 9 - Math SL HB Sem 2

The value of the derivative at a particular point on a curve

Consider a general firnction 5-f(x), a fixed point A(a f(a)) and a variable point
B(x, (x)).
.l: v = l{:r)
/(.0 (...,r(.o)

f(a)

/(.o
a _r
rangent at A with slope /'(a)
f
.f (x)- (a)
The slope ofchord AB =
x-a
Now as B -> A, x -+ a and the slope of chord AB -+ slope oftangent at A

f at: lim "f (x) - "f (a) is the slope of the tangent at x:a and is called
x--ra x-a
the derivative at x:a.
Note :
The slope of the tangent at r:a is defined as the slope of the curve at the point
where x:a, and is the instantaneous rate of change in y with respect to x ai that
point.

Finding the slope using the limit method is said to be using lirst principles.

Example 3:
n f(x) 1 fina (:) by using first principles.
/
x2+l'

The derivative ofa function f can be interpreted two ways:

a) Geometric interpretation of the derivative.
/' is the fi.rnction whose value at x is slope of the tangent line to the graph of f at
x.