Suppose you have a function f such that f(x)=a(b(c(d(x)))) Where a, b, c, and d are all differentiable functions. Asked to find f′(x).

Addison Parker

Addison Parker

Answered question

2022-09-09

Suppose you have a function f such that
f ( x ) = a ( b ( c ( d ( x ) ) ) )
Where a, b, c, and d are all differentiable functions. Asked to find f ( x ).
Can you "chain" the Chain Rule for 4 functions and state that
f ( x ) = a ( b ( c ( d ( x ) ) ) ) b ( c ( d ( x ) ) ) c ( d ( x ) ) d ( x )
And even more generally, can you apply the same pattern for n functions?

Answer & Explanation

Brooklynn Valencia

Brooklynn Valencia

Beginner2022-09-10Added 18 answers

With f = a b c d you have using chain rule:
f ( x ) = ( a ( b c d ) ) ( x ) = a ( ( b c d ) ( x ) ) × ( b c d ) ( x )
using once again chain rule:
f ( x ) = a ( b ( c ( d ( x ) ) ) ) × ( b ( c d ) ) ( x ) = a ( b ( c ( d ( x ) ) ) ) × [ b ( ( c d ) ( x ) ) × ( c d ) ( x ) ]
and using one last time the chain rule on c d you obtain the result.
Nadia Smith

Nadia Smith

Beginner2022-09-11Added 3 answers

Yes you can inductively apply the chain rule for 4 or more functions.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?