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

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DERIVATIVES OF TRIGONOMETRIC FUNCTIONS. 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.

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Presentation Transcript

DERIVATIVES OF

TRIGONOMETRIC FUNCTIONS

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

are positive so we can write

Divide it by

Take the inverse

By the Squeeze Theorem, we have:

However, the function (sin )/ is an even function.

So, its right and left limits must be equal.

Hence, we have:

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:

For what values of x does the graph of f have a horizontal tangent?

Differentiate

• 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).

It turns out that the derivative of the composite

function f ◦ g is the product of the derivatives of f and g.

This fact is one of the most important of the differentiation rules.

It is called the Chain Rule.

• 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:

Chain Rule

• If is differentiable at and is differentiable at , the composite function defined by is differentiable at andis given by the product:

The Chain Rule can be written either

in the prime notation

or,

if y = f(u) and u = g(x), in Leibniz notation:

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:

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 where

• 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:
• )

example:

Differentiate y = (x3 – 1)100

Taking u= x3 – 1 and y = u100

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

Divide both sides by

The slope of a parametrized curve is given by:

The chain rule enables us to find the slope of parametrically defined curves x = x(t) and y = y(t):