<< Chapter < Page Chapter >> Page >

Using the chain rule on a general cosine function

Find the derivative of h ( x ) = cos ( g ( x ) ) .

Think of h ( x ) = cos ( g ( x ) ) as f ( g ( x ) ) where f ( x ) = cos x . Since f ( x ) = sin x . we have f ( g ( x ) ) = sin ( g ( x ) ) . Then we do the following calculation.

h ( x ) = f ( g ( x ) ) g ( x ) Apply the chain rule. = sin ( g ( x ) ) g ( x ) Substitute f ( g ( x ) ) = sin ( g ( x ) ) .

Thus, the derivative of h ( x ) = cos ( g ( x ) ) is given by h ( x ) = sin ( g ( x ) ) g ( x ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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 ( x ) = cos ( 5 x 2 ) .

Let g ( x ) = 5 x 2 . Then g ( x ) = 10 x . Using the result from the previous example,

h ( x ) = sin ( 5 x 2 ) · 10 x = −10 x sin ( 5 x 2 ) .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Using the chain rule on another trigonometric function

Find the derivative of h ( x ) = sec ( 4 x 5 + 2 x ) .

Apply the chain rule to h ( x ) = sec ( g ( x ) ) to obtain

h ( x ) = sec ( g ( x ) tan ( g ( x ) ) g ( x ) .

In this problem, g ( x ) = 4 x 5 + 2 x , so we have g ( x ) = 20 x 4 + 2 . Therefore, we obtain

h ( x ) = sec ( 4 x 5 + 2 x ) tan ( 4 x 5 + 2 x ) ( 20 x 4 + 2 ) = ( 20 x 4 + 2 ) sec ( 4 x 5 + 2 x ) tan ( 4 x 5 + 2 x ) .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the derivative of h ( x ) = sin ( 7 x + 2 ) .

h ( x ) = 7 cos ( 7 x + 2 )

Got questions? Get instant answers now!

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,

d d x ( sin ( g ( x ) ) = cos ( g ( x ) ) g ( x ) d d x sin u = cos u d u d x d d x ( cos ( g ( x ) ) = sin ( g ( x ) ) g ( x ) d d x cos u = sin u d u d x d d x ( tan ( g ( x ) ) = sec 2 ( g ( x ) ) g ( x ) d d x tan u = sec 2 u d u d x d d x ( cot ( g ( x ) ) = csc 2 ( g ( x ) ) g ( x ) d d x cot u = csc 2 u d u d x d d x ( sec ( g ( x ) ) = sec ( g ( x ) tan ( g ( x ) ) g ( x ) d d x sec u = sec u tan u d u d x d d x ( csc ( g ( x ) ) = csc ( g ( x ) ) cot ( g ( x ) ) g ( x ) d d x csc u = csc u cot u d u d x .

Combining the chain rule with the product rule

Find the derivative of h ( x ) = ( 2 x + 1 ) 5 ( 3 x 2 ) 7 .

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

h ( x ) = d d x ( ( 2 x + 1 ) 5 ) · ( 3 x 2 ) 7 + d d x ( ( 3 x 2 ) 7 ) · ( 2 x + 1 ) 5 Apply the product rule. = 5 ( 2 x + 1 ) 4 · 2 · ( 3 x 2 ) 7 + 7 ( 3 x 2 ) 6 · 3 · ( 2 x + 1 ) 5 Apply the chain rule. = 10 ( 2 x + 1 ) 4 ( 3 x 2 ) 7 + 21 ( 3 x 2 ) 6 ( 2 x + 1 ) 5 Simplify. = ( 2 x + 1 ) 4 ( 3 x 2 ) 6 ( 10 ( 3 x 7 ) + 21 ( 2 x + 1 ) ) Factor out ( 2 x + 1 ) 4 ( 3 x 2 ) 6 . = ( 2 x + 1 ) 4 ( 3 x 2 ) 6 ( 72 x 49 ) Simplify.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the derivative of h ( x ) = x ( 2 x + 3 ) 3 .

h ( x ) = 3 4 x ( 2 x + 3 ) 4

Got questions? Get instant answers now!

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 ( x ) = h ( f ( g ( x ) ) ) .

Then, applying the chain rule once we obtain

k ( x ) = d d x ( h ( f ( g ( x ) ) ) = h ( f ( g ( x ) ) ) · d d x f ( ( g ( x ) ) ) .

Applying the chain rule again, we obtain

Questions & Answers

f(x) =3+8+4
tennesio Reply
d(x)(x)/dx =?
Abdul Reply
scope of a curve
Abraham Reply
check continuty at x=1 when f (x)={x^3 if x <1 -4-x^2 if -1 <and= x <and= 10
Raja Reply
what is the value as sinx
Sudam Reply
f (x)=x3_2x+3,a=3
Bilal Reply
given demand function & cost function. x= 6000 - 30p c= 72000 + 60x . . find the break even price & quantities.
Fiseha Reply
hi guys ....um new here ...integrate my welcome
Asif Reply
An airline sells tickets from Tokyo to Detroit for $1200. There are 500 seats available and a typical flight books 350 seats. For every $10 decrease in price, the airline observes and additional 5 seats sold. (a) What should the fare be to maximize profit? (b) How many passeners would be on board?
Ravendra Reply
I would like to know if there exists a second category of integration by substitution
CHIFUNDO Reply
nth differential cofficient of x×x/(x-1)(x-2)
Abhay Reply
integral of root of sinx cosx
Wedraj Reply
the number of gallons of water in a tank t minutes after the tank has started to drain is Q(t)=200(30-t)^2.how fast is the water running out at the end of 10 minutes?
Purity Reply
why is it that the integral of eudu =eu
Maman Reply
using L hospital rule
Abubakar Reply
Practice Key Terms 1

Get the best Calculus volume 1 course in your pocket!





Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask