# Why Is the Chain Rule Not Used in Differentiating h(x) = 3f(x) + 8g(x)?

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In summary, the given problem does not involve a composition of functions and therefore the chain rule is not applicable. The derivative of h(x) can be calculated using the sum rule and constant multiple rules for derivatives, and from the given information, h'(x) can be easily calculated. It is incorrect to say "h'(x) of h(x)" as it does not have a well-defined meaning.
member 731016
Homework Statement
Relevant Equations
For part(a),

The solution is,

However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule? For example if ##f(x) = \sin(x^2)##

Many thanks!

ChiralSuperfields said:

For part(a),
View attachment 326130
The solution is,
View attachment 326131
However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule? For example if ##f(x) = \sin(x^2)##

Many thanks!
There is no inner function. The chain rule is for a composition of functions, like f(g(x)). That does not appear in this problem. The derivative is with respect to x and both f(x) and g(x) are direct functions of x.

member 731016
ChiralSuperfields said:
However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule?
As already noted, there is no "inner function," but the derivative of h(x) (i.e., h'(x)) requires only the use of the sum rule and constant multiple rules for derivatives. Thus h'(x) = 3f'(x) + 8g'(x). From the given information it's easy to calculate h'(4).

BTW, you don't take "h'(x) of h(x)" similar to what you have in the title. You can find the derivative of h(x) or differentiate h(x).

member 731016

## What is the Chain Rule?

The Chain Rule is a fundamental theorem in calculus used to differentiate compositions of functions. It states that if you have two functions, say f(x) and g(x), and you want to differentiate their composition, h(x) = f(g(x)), the derivative is given by h'(x) = f'(g(x)) * g'(x).

## Why is the Chain Rule not applicable to h(x) = 3f(x) + 8g(x)?

The Chain Rule is not applicable because h(x) = 3f(x) + 8g(x) is a linear combination of functions, not a composition of functions. The Chain Rule is used for compositions where one function is nested inside another, not for sums or scalar multiples of functions.

## What rule is used instead of the Chain Rule for h(x) = 3f(x) + 8g(x)?

Instead of the Chain Rule, the linearity of differentiation is used. This rule states that the derivative of a sum of functions is the sum of their derivatives, and the derivative of a constant times a function is the constant times the derivative of the function. Therefore, the derivative of h(x) = 3f(x) + 8g(x) is h'(x) = 3f'(x) + 8g'(x).

## Can you show the differentiation step-by-step for h(x) = 3f(x) + 8g(x)?

Sure. To differentiate h(x) = 3f(x) + 8g(x):1. Differentiate each term separately: The derivative of 3f(x) is 3f'(x) and the derivative of 8g(x) is 8g'(x).2. Combine the results: h'(x) = 3f'(x) + 8g'(x).

## Is there any situation where the Chain Rule could be used in a function involving f(x) and g(x)?

Yes, the Chain Rule would be used if f(x) and g(x) were composed in some way, for instance, if you had a function like h(x) = f(g(x)) or h(x) = g(f(x)). In such cases, you would need to apply the Chain Rule to find the derivative.

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