When we talk about the function f defined for all real numbers x by
𝑓 𝑥 = sin 𝑥, it is understood that sin 𝑥 means the sine of
the angle whose radian measure is 𝑥.
A similar convention holds for the other trigonometric functions
cos, tan, csc, sec, and cot
𝑓 𝑥 = sin 𝑥
For first quadrant all, sin 𝜃 ,
𝜃 , 𝑎𝑛𝑑 tan 𝜃
are positive so we can write
sin 𝜃 < 𝜃 < tan 𝜃
Divide it by sin 𝜃
𝜃
1
1<
<
sin 𝜃
cos 𝜃
sin 𝜃
Take the inverse 1 >
> cos 𝜃
𝜃
lim 1 = 1
𝜃→0+
&
lim cos 𝜃 = 1
𝜃→0+
sin 𝜃
=1
By the Squeeze Theorem, we have: lim+
𝜃→0
𝜃
However, the function (sin 𝜃)/𝜃 is an even function.
So, its right and left limits must be equal.
Hence, we have:
Example:
𝑦 = 𝑥 2 sin 𝑥
Using the same methods as in the case of
finding derivative of sin 𝑥 , we can prove:
The tangent function can also be differentiated by using the definition
of a derivative.
However, it is easier to use the Quotient Rule together with formulas
for derivatives of sin 𝑥 & cos 𝑥 as follows.
The derivatives of the remaining trigonometric functions — csc, sec,
and cot — can also be found easily using the Quotient Rule.
All together:
Example:
Differentiate
For what values of x does the graph of
f have a horizontal tangent?
Since sec x is never 0, we see that f’(x) = 0 when tan x = 1.
This occurs when x = nπ + π/4, where n is an integer
Example:
Find
Example:
Calculate:
How to differentiate composite function 𝐹 𝑥 =
𝑥2 + 1
The differentiation formulas you learned by now
do not enable you to calculate F’(x).
𝐹 𝑥 =𝑓 𝑔 𝑥
𝑤ℎ𝑒𝑟𝑒
𝑓 𝑥 =
𝑥
𝑔 𝑥 = 𝑥2 + 1
𝐹 𝑥 = 𝑓°𝑔
It turns out that the derivative of the composite
function f ◦ g is the product of the derivatives of f and g.
Proof goes over the head, so forget about that.
This fact is one of the most important of the differentiation rules.
It is called the Chain Rule.
It is convenient if we interpret derivatives as rates of change.
Regard:
𝑑𝑢
as the rate of change of u with respect to x
𝑑𝑥
𝑑𝑦
as the rate of change of y with respect to u
𝑑𝑢
𝑑𝑦
as the rate of change of y with respect to x
𝑑𝑥
If u changes twice as fast as x and
y changes three times as fast as u,
it seems reasonable that y
changes six times as fast as x.
So, we expect that:
Chain Rule
If 𝑔 is differentiable at 𝑥 and 𝑓 is differentiable at 𝑔 𝑥 , the
composite function 𝐹 = 𝑓 ◦ 𝑔 defined by 𝐹 𝑥 = 𝑓 𝑔 𝑥 is
differentiable at 𝑥 and 𝐹’ is given by the product:
𝐹’(𝑥) = 𝑓′ 𝑔(𝑥) • 𝑔’(𝑥)
The Chain Rule can be written either
in the prime notation
𝑓°𝑔 ’(𝑥) = 𝑓′ 𝑔(𝑥) • 𝑔’(𝑥)
or,
if y = f(u) and u = g(x), in Leibniz notation:
𝑑𝑦
𝑑𝑦 𝑑𝑢
=
𝑑𝑥
𝑑𝑢 𝑑𝑥
easy to remember because,
if dy/du and du/dx were quotients,
then we could cancel du.
However, remember: du/dx should not
be thought of as an actual quotient
Let’s go back: How to differentiate composite function 𝐹 𝑥 =
𝑥2 + 1
In order not to make your life too complicated (it is already enough),
we’ll introduce one way that is most common and anyway, everyone
ends up with that one: Leibnitz
Let 𝑦 =
𝑢 where 𝑢 = 𝑥2 + 1
𝑑𝑦
𝑑𝑦 𝑑𝑢
1
1
=
=
2𝑥 =
(2𝑥)
2
𝑑𝑥
𝑑𝑢 𝑑𝑥
2 𝑢
2 𝑥 +1
𝑦′ =
𝑥
𝑥2 + 1
dy/dx refers to the derivative of y when y is
considered as a function of x
(called the derivative of y with respect to x)
dy/du refers to the derivative of y when
considered as a function of u
(the derivative of y with respect to u)
example:
Differentiate:
𝑎.
𝑦 = sin(𝑥2)
𝑦 = 𝑠𝑖𝑛 𝑢
𝑢 = 𝑥2
𝑑𝑦
𝑑𝑦 𝑑𝑢
=
= cos 𝑢
𝑑𝑥
𝑑𝑢 𝑑𝑥
𝑏.
2𝑥 = 2𝑥 𝑐𝑜𝑠 𝑥 2
𝑦 = sin2 𝑥
𝑦 = 𝑢2
𝑢 = sin 𝑥
𝑑𝑦
𝑑𝑦 𝑑𝑢
=
= 2𝑢
𝑑𝑥
𝑑𝑢 𝑑𝑥
cos 𝑥 = 2 sin 𝑥 cos 𝑥
example:
Differentiate y = (x3 – 1)100
Taking u = x3 – 1 and y = u100
𝑑𝑦
𝑑𝑦 𝑑𝑢
=
= 100 𝑥 3 − 1
𝑑𝑥
𝑑𝑢 𝑑𝑥
𝑑𝑦
= 300𝑥 2 𝑥 3 − 1
𝑑𝑥
99
99
3𝑥 2
example:
Find f’ (x) if
First, rewrite f as f(x) = (x2 + x + 1)-1/3
Thus,
example:
Find the derivative of
Combining the Power Rule, Chain
Rule, and Quotient Rule, we get:
example:
Differentiate:
y = (2x + 1)5 (x3 – x + 1)4
In this example, we must use the Product Rule before
using the Chain Rule.
The reason for the name ‘Chain Rule’ becomes clear when we make
a longer chain by adding another link.
Suppose that y = f(u), u = g(x), and x = h(t),
where f, g, and h are differentiable functions,
then, to compute the derivative of y with
respect to t, we use the Chain Rule twice:
example:
Notice that we used the Chain Rule twice.
example:
Differentiate
The chain rule enables us to find the slope of
parametrically defined curves x = x(t) and y = y(t):
dy dy dx


dt dx dt
dy
dt  dy
dx
dx
dt
dx
Divide
sides
Theboth
slope
of aby
parametrized
dt
curve is given by:
dy
dy
 dt
dx
dx
dt
