When we talk about the function f defined for all real numbers x by

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