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

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

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.

#Calculus #Proofs

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered 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"

Math	Appears 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

Wow!

Kulkul Chatterjee - 7 years, 1 month ago

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

Guilherme Dela Corte - 7 years, 2 months ago

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 - 6 years, 11 months ago

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 - 6 years, 11 months ago

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 - 6 years, 11 months ago

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

Calvin Lin Staff - 6 years, 11 months ago

@Calvin Lin – My pleasure!

Hussein Hijazi - 6 years, 11 months ago

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

Hussein Hijazi - 6 years, 11 months ago

Math	Appears 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}`	$\boxed{123}$

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

Comments

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