Problem: Find the slope of the tangent line to the curve of f(x) = x at
the point (1,1) and at the point (a,a ).
68 ▯2013 by Sophie Chrysostomou 3.1 DERIVATIVES
The Tangent Line Problem (revisited)
We now generalize how we ﬁnd the equation of the tangent line to the curve of
the function y = f(x), at the point P(a,f(a)). For h ▯= 0 let Q(a+h,f(a+h))
be on the graph and get the slope of the secant PQ:
Rates of Change
Let y = f(x) be a function. If f(x ) = 1 , and1f(x ) = y ,2the ch2nge in x,
called the increment of x is ∆x = x −x and2the c1rresponding change in
y, called the increment of y is ∆y = y − y = f2x ) 1 f(x ). 2 1
The slope of the secant PQ is also called the average rate of change of y
∆y f(x 2 − f(x ) 1
with respect to x and is: = .
∆x x2− x 1
The instantaneous rate of change of y with respect to x is:
∆y f(x 2 − f(x )1
∆x→0 = x2→x 1 . (2)
∆x x 2 x 1
Substituting a for x a1d a + h for x we o2tain the limit of type (1).
69 ▯2013 by Sophie Chrysostomou THE DERIVATIVE
DERIVATIVE OF f
Limits of the form (1) & (2) arise often and, they are given a special name
DEFINITION: Let y = f(x) be a function deﬁned on an open interval
(c,d) and let a ∈ (c, d).
• We say f is diﬀerentiable at x = a if the limit
def f(x) − f(a) f(a + h) − f(a)
f (a) = lim = lim
x→a x − a h→0 h
• This limit above is called the derivative of f at a and is also denoted by
▯ ▯ ▯
dy ▯ df(x) ▯ d ▯
▯ , ▯ , f(x) ▯ , D fxx)| x=a
dx x=a dx x=a dx x=a
• The derivative of f is the function deﬁned by:
f (x) = lim f(x + h) − f(x) , or f (x) = lim f(x) − f(x ) 1
h→0 h x1→x x − x 1
for all x for which the limit on the right hand side of the equation above
• f is diﬀerentiable on an interval (c, d) [or (c, ∞) or (−∞, d) or (−∞, ∞)],
if f (x) exists for all x in the interval. (Note that all these intervals are
• The tangent line to y = f(x) at the point (a,f(a) has slope f (a) and
y − f(a) = f (a)(x − a).
70 ▯2013 by Sophie Chrysostomou 3.1 DERIVATIVES
EXAMPLE: Find the tangent line at a = 1, of the graph of f(x) = .
x + 3
HOMEWORK: (a) Find the slope of the tangent lines to the graph of
f(x) = x at the points (0,0),(1,1),(3,9),(−1,1),(−3,9).
71 ▯2013 by Sophie Chrysostomou THE DERIVATIVE
EXAMPLE: Let f(x) = x. Find f , the derivative of f, and its domain.
72 ▯2013 by Sophie Chrysostomou 3.1 DERIVATIVES
EXAMPLE: Where is f(x) = |x − 1| diﬀerentiable?
73 ▯2013 by Sophie Chrysostomou THE DERIVATIVE
Points where a function is not diﬀerentiable.
Case 1. “Corner”
Case 2. Discontinuity
Case 3. Vertical Tangent
74 ▯2013 by Sophie Chrysostomou 3.1 DERIVATIVES
Theorem : If f is diﬀerentiable at a, then f is continuous at a.
(Note: The converse is false. Functions can be continuous but not diﬀeren-
tiable at a point a.)
Higher Derivatives: If f is diﬀerentiable, then its derivative f is a function.
If f is diﬀerentiable then its derivative is denoted by f and is the second
derivative of f. Successively we may get the the third, fourth ..., derivatives
of f. More notations for higher derivatives are:
2 3 2
d y = d (dy ), d f = d (dy ), f (30(x)▯▯▯ and so on.
dx2 dx dx dx3 dx dx
75 ▯2013 by Sophie Chrysostomou THE DERIVATIVE
3.2 RULES FOR DIFFERENTIATION
1. Derivative of a constant function: If f(x) = c for some constant c, then
f (x) = 0.
2. Power Rule: If f(x) = x , then f (x) = nx n−1 for any nonzero integer
Proof: For positive integers only.
76 ▯2013 by Sophie Chrysostomou 3.2 RULES FOR DIFFERENTIATION
3. Linear Combination Rule: If f and g are diﬀerentiable at x ∈ R and if
r,s ∈ R are constants, then rf + sg is diﬀerentiable at x and
′ ′ ′
(rf + sg) (x) = rf (x) + sg (x)
At this point using rules 1-3 we can ﬁnd the derivative of any polynomial:
4. Product Rule: If f and g are diﬀerentiable at x ∈ R then f ▯ g is diﬀer-
entiable at x and
(f ▯ g) (x) = f (x)g(x) + f(x)g (x) ′
77 ▯2013 by Sophie Chrysostomou THE DERIVATIVE
5. Quotient Rule: If h(x) = g(x) and f and g are diﬀerentiable at x and
g(x) ▯= 0 then h is diﬀerentiable at x and:
h (x) = f (x)g(x) − g (x)f(x) .
The proof is left as homework!
6. Chain Rule: If g is diﬀerentiable at x, and f is diﬀerentiable at g(x),
then f ◦ g is diﬀerentiable at x and
(f ◦ g) (x) = f (g(x)) g (x)
on in Leibniz notation for y(u(x)):
dy dy du
dx du dx
EXAMPLE: Let r(x) = ( x ) . Find r (x) for x ▯= 1.