The generalized power rule says that if g(x) is a function...

The generalized power rule says that if $g(x)$ is a function of $x$ and $y=[g(x)]^{n}$ for any real number $n,$ then
$$\frac{d y}{d x}=n \cdot[g(x)]^{n-1} \cdot g^{\prime}(x)$$
Explain why the generalized power rule is a consequence of the chain rule and the power rule.

Key Concepts

Generalized Power Rule

The generalized power rule combines the ideas of the power rule and the chain rule to handle cases where the base of the power is not simply x but another function of x. When differentiating a function of the form [g(x)]^n, the power rule is applied to the outer exponent, yielding n[g(x)]^(n-1), and then the chain rule is used to multiply by the derivative of the inner function, g'(x). This fusion of the two rules explains why we multiply by g'(x) after applying the conventional power rule.

Chain Rule

The chain rule is a fundamental principle in differentiation that allows us to compute the derivative of composite functions. When a function is composed of an outer function and an inner function, the chain rule states that its derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This rule is essential for understanding how changes in one variable affect another when they are related through a composition of functions.

Power Rule

The power rule is a basic differentiation rule that states that the derivative of a function in the form x^n, where n is any real number, is n multiplied by x raised to the power of (n - 1). This rule simplifies the process of differentiating polynomial functions and other expressions involving exponentiation, making it a core concept in calculus.

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