# 3.6 The chain rule  (Page 3/6)

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## Using the chain rule on a general cosine function

Find the derivative of $h\left(x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right).$

Think of $h\left(x\right)=\text{cos}\left(g\left(x\right)\right)$ as $f\left(g\left(x\right)\right)$ where $f\left(x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}x.$ Since ${f}^{\prime }\left(x\right)=\text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}x.$ we have ${f}^{\prime }\left(g\left(x\right)\right)=\text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right).$ Then we do the following calculation.

$\begin{array}{ccccc}\hfill {h}^{\prime }\left(x\right)& ={f}^{\prime }\left(g\left(x\right)\right){g}^{\prime }\left(x\right)\hfill & & & \text{Apply the chain rule.}\hfill \\ & =\text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right){g}^{\prime }\left(x\right)\hfill & & & \text{Substitute}\phantom{\rule{0.2em}{0ex}}{f}^{\prime }\left(g\left(x\right)\right)=\text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right).\hfill \end{array}$

Thus, the derivative of $h\left(x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right)$ is given by ${h}^{\prime }\left(x\right)=\text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right){g}^{\prime }\left(x\right).$

In the following example we apply the rule that we have just derived.

## Using the chain rule on a cosine function

Find the derivative of $h\left(x\right)=\text{cos}\phantom{\rule{0.1em}{0ex}}\left(5{x}^{2}\right).$

Let $g\left(x\right)=5{x}^{2}.$ Then ${g}^{\prime }\left(x\right)=10x.$ Using the result from the previous example,

$\begin{array}{cc}\hfill {h}^{\prime }\left(x\right)& =\text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(5{x}^{2}\right)·10x\hfill \\ & =-10x\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(5{x}^{2}\right).\hfill \end{array}$

## Using the chain rule on another trigonometric function

Find the derivative of $h\left(x\right)=\text{sec}\phantom{\rule{0.1em}{0ex}}\left(4{x}^{5}+2x\right).$

Apply the chain rule to $h\left(x\right)=\text{sec}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right)$ to obtain

${h}^{\prime }\left(x\right)=\text{sec}\left(g\left(x\right)\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right){g}^{\prime }\left(x\right).$

In this problem, $g\left(x\right)=4{x}^{5}+2x,$ so we have ${g}^{\prime }\left(x\right)=20{x}^{4}+2.$ Therefore, we obtain

$\begin{array}{cc}\hfill {h}^{\prime }\left(x\right)& =\text{sec}\phantom{\rule{0.1em}{0ex}}\left(4{x}^{5}+2x\right)\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}\left(4{x}^{5}+2x\right)\left(20{x}^{4}+2\right)\hfill \\ & =\left(20{x}^{4}+2\right)\text{sec}\phantom{\rule{0.1em}{0ex}}\left(4{x}^{5}+2x\right)\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}\left(4{x}^{5}+2x\right).\hfill \end{array}$

Find the derivative of $h\left(x\right)=\text{sin}\left(7x+2\right).$

${h}^{\prime }\left(x\right)=7\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(7x+2\right)$

At this point we provide a list of derivative formulas that may be obtained by applying the chain rule in conjunction with the formulas for derivatives of trigonometric functions. Their derivations are similar to those used in [link] and [link] . For convenience, formulas are also given in Leibniz’s notation, which some students find easier to remember. (We discuss the chain rule using Leibniz’s notation at the end of this section.) It is not absolutely necessary to memorize these as separate formulas as they are all applications of the chain rule to previously learned formulas.

## Using the chain rule with trigonometric functions

For all values of $x$ for which the derivative is defined,

$\begin{array}{cccc}\frac{d}{dx}\left(\text{sin}\left(g\left(x\right)\right)\phantom{\rule{0.2em}{0ex}}=\text{cos}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right)g\prime \left(x\right)\hfill & & & \frac{d}{dx}\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.2em}{0ex}}=\text{cos}\phantom{\rule{0.1em}{0ex}}u\frac{du}{dx}\hfill \\ \frac{d}{dx}\left(\text{cos}\left(g\left(x\right)\right)\phantom{\rule{0.1em}{0ex}}=\text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right)g\prime \left(x\right)\hfill & & & \frac{d}{dx}\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.1em}{0ex}}=\text{−}\text{sin}\phantom{\rule{0.1em}{0ex}}u\frac{du}{dx}\hfill \\ \frac{d}{dx}\left(\text{tan}\left(g\left(x\right)\right)\phantom{\rule{0.22em}{0ex}}={\text{sec}}^{2}\left(g\left(x\right)\right)g\prime \left(x\right)\hfill & & & \frac{d}{dx}\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.2em}{0ex}}={\text{sec}}^{2}u\frac{du}{dx}\hfill \\ \frac{d}{dx}\left(\text{cot}\left(g\left(x\right)\right)\phantom{\rule{0.25em}{0ex}}=\text{−}{\text{csc}}^{2}\left(g\left(x\right)\right)g\prime \left(x\right)\hfill & & & \frac{d}{dx}\phantom{\rule{0.1em}{0ex}}\text{cot}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.25em}{0ex}}=\text{−}{\text{csc}}^{2}u\frac{du}{dx}\hfill \\ \frac{d}{dx}\left(\text{sec}\left(g\left(x\right)\right)\phantom{\rule{0.25em}{0ex}}=\text{sec}\left(g\left(x\right)\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right)g\prime \left(x\right)\hfill & & & \frac{d}{dx}\phantom{\rule{0.1em}{0ex}}\text{sec}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.25em}{0ex}}=\text{sec}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}u\frac{du}{dx}\hfill \\ \frac{d}{dx}\left(\text{csc}\left(g\left(x\right)\right)\phantom{\rule{0.25em}{0ex}}=\text{−}\text{csc}\left(g\left(x\right)\right)\text{cot}\phantom{\rule{0.1em}{0ex}}\left(g\left(x\right)\right)g\prime \left(x\right)\hfill & & & \frac{d}{dx}\phantom{\rule{0.1em}{0ex}}\text{csc}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.25em}{0ex}}=\text{−}\text{csc}\phantom{\rule{0.1em}{0ex}}u\phantom{\rule{0.1em}{0ex}}\text{cot}\phantom{\rule{0.1em}{0ex}}u\frac{du}{dx}.\hfill \end{array}$

## Combining the chain rule with the product rule

Find the derivative of $h\left(x\right)={\left(2x+1\right)}^{5}{\left(3x-2\right)}^{7}.$

First apply the product rule, then apply the chain rule to each term of the product.

$\begin{array}{ccccc}\hfill {h}^{\prime }\left(x\right)& =\frac{d}{dx}\left({\left(2x+1\right)}^{5}\right)·{\left(3x-2\right)}^{7}+\frac{d}{dx}\left({\left(3x-2\right)}^{7}\right)·{\left(2x+1\right)}^{5}\hfill & & & \text{Apply the product rule.}\hfill \\ & =5{\left(2x+1\right)}^{4}·2·{\left(3x-2\right)}^{7}+7{\left(3x-2\right)}^{6}·3·{\left(2x+1\right)}^{5}\hfill & & & \text{Apply the chain rule.}\hfill \\ & =10{\left(2x+1\right)}^{4}{\left(3x-2\right)}^{7}+21{\left(3x-2\right)}^{6}{\left(2x+1\right)}^{5}\hfill & & & \text{Simplify.}\hfill \\ & ={\left(2x+1\right)}^{4}{\left(3x-2\right)}^{6}\left(10\left(3x-7\right)+21\left(2x+1\right)\right)\hfill & & & \text{Factor out}\phantom{\rule{0.2em}{0ex}}{\left(2x+1\right)}^{4}{\left(3x-2\right)}^{6}.\hfill \\ & ={\left(2x+1\right)}^{4}{\left(3x-2\right)}^{6}\left(72x-49\right)\hfill & & & \text{Simplify.}\hfill \end{array}$

Find the derivative of $h\left(x\right)=\frac{x}{{\left(2x+3\right)}^{3}}.$

${h}^{\prime }\left(x\right)=\frac{3-4x}{{\left(2x+3\right)}^{4}}$

## Composites of three or more functions

We can now combine the chain rule with other rules for differentiating functions, but when we are differentiating the composition of three or more functions, we need to apply the chain rule more than once. If we look at this situation in general terms, we can generate a formula, but we do not need to remember it, as we can simply apply the chain rule multiple times.

In general terms, first we let

$k\left(x\right)=h\left(f\left(g\left(x\right)\right)\right).$

Then, applying the chain rule once we obtain

${k}^{\prime }\left(x\right)=\frac{d}{dx}\left(h\left(f\left(g\left(x\right)\right)\right)=h\prime \left(f\left(g\left(x\right)\right)\right)·\frac{d}{dx}f\left(\left(g\left(x\right)\right)\right).$

Applying the chain rule again, we obtain

why n does not equal -1
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
y=800
800
Bg
how do u factor the numerator?
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
what is a function? f(x)
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
is x=2 a function?
The
What is limit
it's the value a function will take while approaching a particular value
Dan
don ger it
Jeremy
what is a limit?
Dlamini
it is the value the function approaches as the input approaches that value.
Andrew
Thanx
Dlamini
Its' complex a limit It's a metrical and topological natural question... approaching means nothing in math
Antonio
is x=2 a function?
The
3y^2*y' + 2xy^3 + 3y^2y'x^2 = 0 sub in x = 2, and y = 1, isolate y'
what is implicit of y³+x²y³=5 at (2,1)
tel mi about a function. what is it?
Jeremy
A function it's a law, that for each value in the domaon associate a single one in the codomain
Antonio
function is a something which another thing depends upon to take place. Example A son depends on his father. meaning here is the father is function of the son. let the father be y and the son be x. the we say F(X)=Y.
Bg
yes the son on his father
pascal
a function is equivalent to a machine. this machine makes x to create y. thus, y is dependent upon x to be produced. note x is an independent variable
moe
x or y those not matter is just to represent.
Bg