Differential Calculus - Chain Rule for Trigonometric Functions

Summary

This video covers the chain rule in differential calculus, specifically focusing on trigonometric functions. It explains how to differentiate composite trigonometric functions like sine, cosine, tangent, secant, cotangent, and cosecant using the chain rule. The instructor walks through several examples, highlighting common mistakes and emphasizing the importance of parentheses in simplifying the results. By the end, viewers learn how to differentiate functions within functions, apply the chain rule, and simplify expressions accurately. The video provides clear explanations for students to better understand and apply these concepts.

Takeaways

• 📘 The video covers the chain rule for trigonometric functions in differential calculus.
• 📌 The derivative of sin(U) is equivalent to cos(U) multiplied by U'.
• 🔍 The derivative of tan(U) is sec^2(U) multiplied by U'.
• ✏️ The derivative of cos(U) is -sin(U) multiplied by U'.
• 📐 For cot(U), the derivative is -csc^2(U) multiplied by U'.
• 📈 The chain rule is applied by differentiating the outer function, then multiplying by the derivative of the inner function (U').
• 📝 Example 1: For y = sin(2x), the derivative is 2 cos(2x).
• 🔑 Example 2: For y = cos(x-1), the derivative is -sin(x-1).
• 📊 Example 3: For y = tan(3x), the derivative is 3 sec^2(3x).
• ✅ The key takeaway is understanding how to apply the chain rule effectively for differentiating composite trigonometric functions.

Q & A

What is the chain rule in the context of trigonometric functions?

The chain rule for trigonometric functions involves differentiating the outer function normally and then multiplying by the derivative of the inner function (U), which is referred to as U prime (U').

What is the derivative of sin(U) using the chain rule?

The derivative of sin(U) using the chain rule is cos(U) multiplied by U'.

How do you apply the chain rule to differentiate tan(U)?

To differentiate tan(U) using the chain rule, the result is sec²(U) multiplied by U'.

What is the formula for differentiating sec(U) using the chain rule?

The derivative of sec(U) using the chain rule is sec(U)tan(U) multiplied by U'.

What happens when you differentiate cos(U) using the chain rule?

When differentiating cos(U) using the chain rule, the result is -sin(U) multiplied by U'.

How do you find the derivative of cot(U) using the chain rule?

The derivative of cot(U) using the chain rule is -csc²(U) multiplied by U'.

What steps should be followed when applying the chain rule to differentiate y = sin(2x)?

First, identify the inner function U = 2x. Then, differentiate sin(U) to get cos(2x) and multiply it by the derivative of U, which is 2. The final answer is 2cos(2x).

How do you apply the chain rule to y = cos(x - 1)?

First, identify U = x - 1. Then, differentiate cos(U) to get -sin(x - 1) and multiply it by the derivative of U, which is 1. The final answer is -sin(x - 1).

What is the result of differentiating y = tan(3x) using the chain rule?

First, identify U = 3x. Differentiate tan(U) to get sec²(3x) and multiply it by the derivative of U, which is 3. The final answer is 3sec²(3x).

How do you differentiate y = sec(5x) using the chain rule?

First, identify U = 5x. Differentiate sec(U) to get sec(5x)tan(5x) and multiply by the derivative of U, which is 5. The final answer is 5sec(5x)tan(5x).

Outlines

00:00

📘 Introduction to Chain Rule for Trigonometric Functions

The paragraph introduces the concept of the chain rule in differential calculus, focusing on trigonometric functions. It lists the derivatives of the six trigonometric functions (sin, tan, sec, cos, cot, and csc) with respect to an internal function 'U'. The method involves differentiating the outer function normally and then multiplying by the derivative of the inner function (U'). The example provided is y = sin(2x), demonstrating the chain rule process to obtain the final derivative.

05:01

📊 Example: Differentiating a Composite Cosine Function

This paragraph provides a second example where y = cos(x - 1). The chain rule is applied to differentiate the cosine function, with U being (x - 1). The derivative of y is calculated using the formula for the derivative of cosine, resulting in the final answer. Special emphasis is placed on correctly applying parentheses to avoid mistakes when writing the final expression.

10:03

🧮 Example: Derivative of Tangent Function with Inner Function

This example deals with y = tan(3x). The paragraph explains how to identify U as 3x and then apply the chain rule, using the formula for the derivative of tangent (sec²U). The derivative of y is simplified to the final form, demonstrating the application of the chain rule in trigonometric functions. Attention is given to proper simplification of terms involving sec².

15:05

🧑‍🏫 Applying Sum and Difference Rule in Chain Rule

This section introduces the sum and difference rule in the context of chain rule differentiation, using the example y = csc(5x) + 5. The csc(5x) part is differentiated separately, and then the sum rule is applied. The final derivative involves multiplying the derivative of csc by cotangent and simplifying to get the final result. The section also reiterates the importance of using parentheses correctly.

20:07

📐 Chain Rule in Complex Cosine Function Differentiation

The paragraph focuses on examples involving powers of trigonometric functions, specifically dealing with y = cos(3x²). The chain rule is applied by differentiating the outer cosine function and then the inner 3x² function. The final result is simplified, demonstrating the step-by-step application of the rule. The concept of treating cosine of constants differently is also highlighted in this section.

Keywords

💡Chain Rule

The chain rule is a fundamental rule in calculus used to differentiate composite functions. It states that if you have a function within a function, you must first differentiate the outer function and then multiply it by the derivative of the inner function (U'). In the video, this rule is applied to trigonometric functions, demonstrating how to differentiate expressions like sin(2x) and tan(3x).

💡Trigonometric Functions

Trigonometric functions include sine (sin), cosine (cos), tangent (tan), and others. These functions are commonly used in calculus to describe relationships in geometry, especially with circles and angles. In the video, the derivatives of the six primary trigonometric functions are used to demonstrate how the chain rule is applied to differentiate composite trigonometric functions.

💡Sine Function (sin)

The sine function (sin) is a trigonometric function that represents the y-coordinate of a point on the unit circle as an angle varies. In the context of the video, the derivative of sin(u) is explained using the chain rule, where the outer function (sin) is differentiated first, and then multiplied by the derivative of the inner function (U'). For example, y = sin(2x) leads to y' = 2cos(2x).

💡Cosine Function (cos)

The cosine function (cos) represents the x-coordinate of a point on the unit circle as an angle varies. In the video, the derivative of cos(u) is given as -sin(u) multiplied by U'. This rule is applied in examples such as differentiating y = cos(x - 1), where y' = -sin(x - 1).

💡Tangent Function (tan)

The tangent function (tan) is the ratio of the sine to the cosine of an angle. In the video, the derivative of tan(u) is shown to be sec²(u) multiplied by U'. This is demonstrated in an example where y = tan(3x), leading to y' = 3sec²(3x).

💡Secant Function (sec)

The secant function (sec) is the reciprocal of the cosine function. The video shows that the derivative of sec(u) is sec(u)tan(u) multiplied by U'. This concept is applied when differentiating more complex trigonometric expressions involving secant functions.

💡Composite Function

A composite function is a function made by combining two or more functions, where one function is applied to the result of another. The video focuses on differentiating composite trigonometric functions using the chain rule. For instance, sin(2x) is a composite function because 2x is inside the sine function, requiring the use of the chain rule to differentiate it.

💡U-Prime (U')

U' refers to the derivative of the inner function in a composite function. In the chain rule, after differentiating the outer function, you multiply the result by U', which represents the derivative of the inner function. The video demonstrates this process by finding U' for various trigonometric examples, such as y = sin(2x), where U' = 2.

💡Negative Sine (-sin)

The derivative of the cosine function is negative sine (-sin). This concept is emphasized in the video as part of the chain rule when differentiating composite functions involving cosine. For example, y = cos(3x) results in y' = -3sin(3x), showing how the negative sign is essential in the differentiation process.

💡Sum and Difference Rule

The sum and difference rule in calculus states that the derivative of a sum or difference of two functions is the sum or difference of their derivatives. In the video, this rule is applied when differentiating expressions like y = cot(5x) + 5. Each part is differentiated separately, and then the results are combined.

Highlights

Introduction to the chain rule for trigonometric functions, focusing on differentiation in calculus.

The derivative of sin(u) is presented as sin(u) * u', emphasizing the chain rule.

For tan(u), the derivative is derived as sec^2(u) * u'.

For sec(u), the derivative is given as sec(u) * tan(u) * u'.

The derivative of cos(u) is expressed as -sin(u) * u', showing the application of the chain rule.

Derivatives of cot(u) and csc(u) are discussed, highlighting their negative derivatives.

In the first example, the chain rule is applied to sin(2x), and the derivative is computed as 2 * cos(2x).

In the second example, the chain rule is used for cos(x - 1), resulting in -sin(x - 1).

For the third example, tan(3x) is differentiated as sec^2(3x) * 3, emphasizing the chain rule.

An example involving cot(5x) + 5 demonstrates the use of sum and difference rules with chain rule application.

Clarification on the importance of open and close parentheses in differentiating trigonometric functions.

The chain rule applied to complex trigonometric functions like cos(3x^2), showing detailed differentiation steps.

Using constant multiple rule for functions like cos(3) * x^2, showing how constants are treated in differentiation.

Differentiation of square roots in trigonometric functions, such as sqrt(cos(x)), by rewriting them in exponent form.

Conclusion: Emphasizing the importance of open and close parentheses, constant multiple rule, and chain rule when differentiating trigonometric functions.

Transcripts

hello class for this

video chain rule for trigonometric

functions this is a topic in

differential calculus the chain rule

versions of the derivatives of the six

trigonometric functions are shown below

first one we have the derivative of sin

U is equivalent to cine u u Prime second

we have the derivative of tan U is

equivalent to Second s u u Prime third

we have the derivative of second U U is

equivalent to Second U tangent u u Prime

fourth one we have the derivative of

cosine U is equivalent to netive sin u u

Prime next we have the derivative of

cotangent U is equivalent to netive cant

s u u Prime and last one we have the

derivative of cant U is equivalent to

negative cyant ENT u u

[Music]

Prime chain rule trigonometric function

all we need to do is to differentiate

them normally and then ilag

derivative U which is our U primea we

are differentiating a function within

our

function on how to apply chain rule on

our trigonometric functions first one we

have y is equivalent to S of 2x so as

you can observe class we have another

function within our function because you

X is mying two so this is another

function within our trigonometric

function so we have our formula Kina the

derivative of our sin U is equivalent to

cine u u Prime so U is equivalent to 2x

next thing we will derive our U that is

our U Prime is equivalent to two because

if we are deriving X raed to one that is

equivalent to one so u u Prime so proed

in differentiating our y that is

equivalent to Y Prime equals sign ising

cosine so that is cosine of is

substitute that is 2x multipied by our U

Prime which is two so copy paste two so

different shaping all we need to do here

is to simplify that is y Prime is

equivalent to take note multiply because

this is a trigonometric function

so

is front that is two cosine of 2x so

simplify PA and I think Hindi so this

will be our final answer so class

pite to avoid confusion that is 2 cosine

of open parentheses of 2x close

parentheses emphasize 2x is cosine so

this is our final answer so

P for our second example we have y is

equivalent to cosine of open parentheses

of x - 1 closed parenthesis so we will

be using this formula we have derivative

of cosine U is equivalent to negative of

sin U U Prime first thing that we need

to do is to determine U which

is cosine function which is our xus

one and then derive u u Prime that is

equivalent to the derivative of x is 1

the derivative of1 is zero so u u Prime

so let us proceed in deriving our y that

is y Prime is equivalent

toag cosine U is negative sin U so

negative open parenthesis of s then U is

xus one so open parentheses x

-1 close parenthesis close parenthesis

next thing is U Prime so multiply U

Prime which is one so we're done in

differentiating all we need to do is to

simplify this one that is equivalent to

negative imultiply

s of

xus so this will be our final

answer simplify common mistake class Isa

open and close parenthesis for example

negative s of x -1 so if this one

isare this is not equal

because is sin and then is deded by

one

avidus open and

clthes open and cl parentheses emphasize

xus one is the function of our sign so

first

example operation between

so open and close

parenthesis second example is

Operation function so again

class Final Answer third example is y is

equivalent to tangent of 3x first thing

that we need to do is to

determine which

isent which is c3x next class is to

derive our U which is our U Prime the

derivative of 3x is simply three so we

will be using this formula the

derivative of tangent U is equivalent to

Second s u u Prime so derive that is y

Prime is equivalent to tan U is second

squar of our U which is our U is

3X 3X

open and close parenthesis multiplied by

our U Prime which is three so we're done

in differentiating all we need to do is

to simplify so take note Class B IM

multiply because we are dealing with

trigonometric

function

so second squar of 3x and this will be

our final answer so take note class open

and close

parentheses

oper but again class to avoid

mes open and

cl trigonometric function

so Final Answer letter D or our fourth

example we have the function of X is

equivalent to open parenthesis of cant

5X close parentheses + 5 so clarate so

we will be using sum and difference rule

we just need to differentiate them

individually so for our first

part formula which is the derivative of

cyant U is equivalent to negative open

parentheses of cyant U cotangent U close

parentheses U Prime so first thing that

we need to do is to identify our U that

is equivalent

to cose function which is 5X next class

the derivative of our U is equivalent to

5 so proceed put first part which is the

derivative of our cose 5x we have F

Prime of X is equivalent to the

derivative of cant 5x is equivalent to

negative open parenthesis of cose of u u

is 5X lagay

cotangent of U so U is another

5x close parenthesis close parenthesis

then U Prime not in class is five

so part proceed second part which is the

derivative of our five so as we all know

the derivative with respect to X of any

constant is equivalent to zero so this

is plus Z so we're done differentiating

all we need to do is to simplify so fime

of X is equivalent to again class

multiply is front so

isant of 5x cotangent of

5x and this will be our final answer so

P Wala open and close parenthesis okay

Lang but again to avoid

mistakes that

is5 cose of open parentheses 5x

multiplied by cotangent of open and Clos

parentheses 5x so this will also be our

final answer so class to avoid confusion

open and close parenthesis here are our

examples first one is y is equivalent to

cosine of 3x^2 second we have y is

equivalent to open parentheses cosine of

3 Clos parentheses x^2 third we have y

is equivalent to cosine open parentheses

of 3x close parentheses squar fourth we

have y is equivalent to cosine 2 x and

last one we have y is equivalent to the

square root of cosine of x soive class

and let us

determine is the derivative of our

cosine U is equivalent to netive sin u u

Prime let's start with our letter A that

is equivalent to cosine of open

parenthesis

3x s so same open and

clthes

3X and cl parenthesis which is

3x² and then U Prime is equivalent to 3

* 2 is 6 x then 2 - 1 is 1 so 6 x to 1

so apply class formula y Prime is

equivalent to negative open parenthesis

cosine is mag

s then copy paste C that is 3x squared

close parenthesis then multiplied by our

U Prime so to U Prime that is

6X last thing that we need to do is

simplify so again Class

B

so

6X s of

3x^2 and this will be our final answer

so open and close parentheses 3x2 that

is s of open parenthesis 3 x² closed

parenthesis final answer so how about if

separate cosine 3 then open and close

parenthesis tapos multipli by X so analy

class cosine of 3 is a constant

variable so we will just treat cosine 3

as a

constant we will just differentiate this

one using our constant multiple rule so

recall the derivative with respect to X

of a constant u n is equivalent to

constant n u n minus one so y Prime is

equivalent to our constant which is a

cosine of

3 multiplied by our n which is

squared multiplied by our U which is our

X then 2 - 1 is one so rearrange class

this is equivalent to 2x cosine of 3

this is our final answer so

again

is so final answer

so front for our letter C we have cosine

of open parenthesis 3x then close

parenthesis squared so squ

CL is

3x

soite that is equivalent to cosine of 3^

s will be 9 then X is squared so that is

equivalent to x^ 2 so cosine of 9 x^2 is

same cosine of open parenthesis 9 x^2

closed parenthesis so we will be using

this formula the derivative of cosine of

U is equivalent to Nega of s u u Prime

first thing that we need to do is to

identify our U which is equivalent to

which is 9 X2 next one derive u u Prime

is equivalent to 9ti by 2 is 18 and then

x 2 - 1 is 1 so raised to 1 so

gam constant multiple rule so class we

have our U we have our U Prime so

proceed to in differentiating that is y

Prime is equivalent

to that is negative then open

parenthesis cosine is magig s u so magig

s of 9 X2 close parenthesis U Prime

class is C

18 x so as I have said

earlier is ilagay front so that is

equivalent to

-8x sin of 9 X2 this will be our final

answer so para sure

P open and close parentheses 9x2 so s of

open parentheses of 9 x^2 close

parenthesis final answer

so

okay

pro d that is

yal that is just equivalent to open

parenthesis of cosine X Clos parentheses

squ Sil is squared so as per

observation is chain rule constant

multiple

rule mle

rule U Prime because we are

differentiating a function within a

function so identify First what is our c

that

isan Conant front which is exponent that

is cosine of x

proceed U Prime which is the derivative

of our U so as we all know the

derivative with respect to X of our

cosine of x is equivalent to negative s

of X so the derivative of our cosine of

x is negative sin of X last one is

exponent which is n that is equivalent

to

two u u Prime and N let's proceed in

differentiating the is equivalent

to formula constant is n is two U is

cosine of x and then n is 2us 1 is 1

then U Prime is

netive sin of X take note open and close

parenthesis

so so that is equivalent

to negative so Lang front

negative -2 cosine of x sin of

X and this will be our final answer

P we have NE -2 sin of X and cosine of x

that is just the same so that is

equivalent to -2 sin of X multiplied by

cosine of x so same

Lang last example class is y is

equivalent to the square root of our

cosine of x so we cannot differentiate

this immediately we need to rewrite this

one that is equivalent to open

parenthesis cosine of x then rais to the

power of 12 because as we all know is

power of one square root is two so we

will be using end root of X ra to m is

equivalent X to m / n

soite and proed

inting Rule Conant multiple rule which

is this one so

identify front next

one U which is exponent which is cosine

of X

proceed U Prime which is the derivative

of cosine

X which is netive

sinx last thing is our exponent which is

n that is equivalent to 12 so class we

have our constant u u Prime and our n so

proceed differentiating y Prime is

equivalent to constant is w and

is

12 U class is cosine of x and then

exponent is 12 minus one is

one2 calculator para sure and last thing

is your U Prime which is negative sin of

X so in differentiating last thing that

we need to do is simplify

soay negative it transfer s front

12 of cosine of x raised to the power

of2 multiplied by

sinx so Final Answer exponent negative

so it

transfera so in transferring class

always remember that X ra to m is

equivalent to 1 /x to the power of M so

transfer class

negative sin

xang that is sin x divided by our cosine

of x then raised to the power of POS 12

and then

class that is two so

adjust so as per

observation okay Lang radical sa Baba so

this is our final answer so class if

choices you just need to

rationalize okay final answer so that

ends our topic for this video class do

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