Waste less time on Facebook — follow Brilliant.
×

\( (f \circ g) ' = f' \circ g + f \circ g ' \)

Sometimes, we can confuse applying the chain rule with applying the product rule.

The product rule states that \( [ f(x) \times g(x) ]' = f'(x) \times g(x) + f(x) \times g'(x), \) while the chain rule states that \( (f \circ g )' (x) = g'(x) \times f' \circ g (x) \).

Find infinitely many pairs of functions such that

\( (f \circ g) ' = f' \circ g + f \circ g ' \)

Example: \( f(x) = e^{-x} \) and \( g(x) = 1 + x - e^{-x} \).


This is a list of Calculus proof based problems that I like. Please avoid posting complete solutions, so that others can work on it.

Note by Calvin Lin
3 years, 8 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

Wow!

Kulkul Chatterjee - 3 years, 7 months ago

Log in to reply

Hint: \(f(x) = a(x), \; g(x) = b(x) - a(x)\)?

Guilherme Dela Corte - 3 years, 8 months ago

Log in to reply

I'm pretty sure you have the rules confused. The first one you listed is the product rule, as it is the multiplcation of two functions. The second one, which has f and g as composite functions, is in fact the chain rule...

Pretty sure thats what going on.

Hussein Hijazi - 3 years, 5 months ago

Log in to reply

As stated, "Sometimes, we can confuse applying the chain rule with applying the product rule."

I am intentionally asking you to find pairs of functions in which the "confused" version ends up being correct. It is certainly not true of any (differentiable) functions \(f\) and \(g\).

Calvin Lin Staff - 3 years, 5 months ago

Log in to reply

Unless what you stated is the perceived "confused" version, then my bad. 'Twas a bit hard to notice

Hussein Hijazi - 3 years, 5 months ago

Log in to reply

I understand that. But in the introduction where you say "The chain rule states that [Latex stuff] while the product rule states that [more Latex stuff]" but those Latex areas should be switched because the chain rule does not state what you said nor does the product rule state what is followed.

I understand some composite functions, when differentiated, turn out to be like the product rule. But what you said in the beginning isn't true.

If it's some kind of humor I'm not seeing, I'm sorry.

Hussein Hijazi - 3 years, 5 months ago

Log in to reply

@Hussein Hijazi I completely missed that. Thanks for pointing it out! I've made the corresponding edits.

Calvin Lin Staff - 3 years, 5 months ago

Log in to reply

@Calvin Lin My pleasure!

Hussein Hijazi - 3 years, 5 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...