Which rule would be used to differentiate the function

Question

BlackTom AI · Accepted Answer

The problem presents a function that is a composition of two functions: the outer function is the cube, and the inner function is the polynomial inside the parentheses.

Because f(x) = (5x^4 − 3x^2 + 7)^3 is of the form [g(u)] with u = 5x^4 − 3x^2 + 7 and g(u) = u^3, differentiation requires the chain rule: derivative of the outer function evaluated at the inner function times the derivative of the inner function.

If we denote u(x) = 5x^4 − 3x^2 + 7, then f(x) = [u(x)]^3. The chain rule gives f'(x) = 3 [u(x)]^2 · u'(x).

Next, compute u'(x) for the inner polynomial: u'(x) = 20x^3 − 6x. So the full derivative is f'(x) = 3 (5x^4 − 3x^2 + 7)^2 · (20x^3 − 6x).

This reasoning aligns with the general idea of the chain rule: when a function is a composition of two functions, differentiate the outer function and multiply by the derivative of the inner function.

Other common differentiation rules (like Product Rule, Quotient Rule, or Special-Case rules) would be unnecessary here because there is no explicit product or quotient of multiple functions outside the nested structure; the entire expression is a single composite function raised to a power.

If the multiple-choice options had included choices like Product Rule, Chain Rule, Quotient Rule, or Limit, the correct choice would be Chain Rule for differentiating a composite function of the form [something]^power.

Which rule would be used to differentiate the function

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