Take the inverse Example: The derivatives of the remaining trigonometric functions — csc, sec, and...
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Transcript of Take the inverse Example: The derivatives of the remaining trigonometric functions — csc, sec, and...
When we talk about the function f defined for all real numbers x by , it is understood that means the sine of the angle whose radian measure is .A similar convention holds for the other trigonometric functions cos, tan, csc, sec, and cot
DERIVATIVES OF
TRIGONOMETRIC
FUNCTIONS
𝑓 (𝑥 )=sin 𝑥
For first quadrant all, are positive so we can write
sin 𝜃<𝜃< tan𝜃
Divide it by 1¿𝜃
sin 𝜃<
1cos𝜃
Take the inverse1¿sin 𝜃𝜃
>cos𝜃
By the Squeeze Theorem, we
have:
lim𝜃→0+¿ sin 𝜃
𝜃=1¿
¿
However, the function (sin )/ is an even function.So, its right and left limits must be equal.Hence, we have:
Example: 𝑦=𝑥2sin 𝑥
Using the same methods as in the case of finding derivative of , 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 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:
CHAIN RULE
How to differentiate composite function
The differentiation formulas you learned by now do not enable you to calculate F’(x).
𝐹 (𝑥 )= 𝑓 [𝑔 (𝑥 ) ] h𝑤 𝑒𝑟𝑒 𝑓 (𝑥 )=√𝑥𝑔 (𝑥 )=𝑥2+1
𝐹 (𝑥 )= 𝑓 °𝑔
It turns out that the derivative of the compositefunction 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:
as the rate of change of u with respect to x
as the rate of change of y with respect to u
as the rate of change of y with respect to x
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:
If is differentiable at and is differentiable at , the composite function defined by is differentiable at and is given by the product:
Chain Rule
The Chain Rule can be written either
in the prime notation
or,
if y = f(u) and u = g(x), in Leibniz notation:
𝑑𝑦𝑑𝑥
=𝑑𝑦𝑑𝑢
𝑑𝑢𝑑𝑥
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
How to differentiate composite function Let’s go back:
Let where
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
𝑑𝑦𝑑𝑥
=𝑑𝑦𝑑𝑢
𝑑𝑢𝑑𝑥
=1
2√𝑢(2 𝑥 )= 1
2√𝑥2+1(2𝑥 )
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)
Differentiate:
)
example:
𝑑𝑦𝑑𝑥
=𝑑𝑦𝑑𝑢
𝑑𝑢𝑑𝑥
= (cos𝑢) (2𝑥 )=2𝑥 𝑐𝑜𝑠 (𝑥2 )
𝑏 . 𝑦=sin 2 𝑥
𝑑𝑦𝑑𝑥
=𝑑𝑦𝑑𝑢
𝑑𝑢𝑑𝑥
= (2𝑢 ) (cos 𝑥 )=2sin𝑥 cos 𝑥
Differentiate y = (x3 – 1)100
example:
Taking u = x3 – 1 and y = u100
𝑑𝑦𝑑𝑥
=𝑑𝑦𝑑𝑢
𝑑𝑢𝑑𝑥
=100 (𝑥3−1 )99 ( 3 𝑥2 )
𝑑𝑦𝑑𝑥
=300 𝑥2 (𝑥3−1 )99
example:
Find f’ (x) if
First, rewrite f as f(x) = (x2 + x + 1)-1/3
Thus,
Find the derivative of
example:
Combining the Power Rule, Chain Rule, and Quotient Rule, we get:
Differentiate: y = (2x + 1)5 (x3 – x + 1)4
In this example, we must use the Product Rule before using the Chain Rule.
example:
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 withrespect to t, we use the Chain Rule twice:
example:
Notice that we used the Chain Rule twice.
Differentiate
example:
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
dydydt
dx dxdt
Divide both sides bydx
dtThe slope of a parametrized curve is given by:
dydy dt
dxdxdt