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

[ChiralSuperfields]

Homework Statement

Please see below

Relevant Equations

Please see below

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!

 

[FactChecker]

Homework Statement: Please see below
Relevant Equations: Please see below

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.

 

[Mark44]

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).