91Ó°ÊÓ

The Generalized Power Rule is an extension of the basic power rule for differentiation, which states that the derivative of \(x^n\) is \(nx^{n-1}\). It allows us to differentiate more complex expressions, especially those in the form of \((g(x))^n\). In the case where you have a function consisting of a power of another function, the generalized power rule combines itself with the use of the chain rule and product rule.

When you have a function like \(f(x) = x^2 \sqrt{1 + x^2}\), it's essential to first rewrite it with power expressions if needed. We express it as \(f(x) = x^2 (1 + x^2)^{1/2}\) to simplify differentiation. Notice that the expression \((1+x^2)^{1/2}\) takes the form \((g(x))^n\), thus allowing us to apply the generalized power rule.

The Product Rule is a fundamental concept in calculus used to differentiate products of two functions. For functions \(u(x)\) and \(v(x)\), the product rule states:

• \( f'(x) = u'(x) v(x) + u(x) v'(x) \)

In our exercise, the function \( f(x) = x^2 \sqrt{1+x^2}\) can be split into two components: \( u(x) = x^2 \) and \( v(x) = (1+x^2)^{1/2} \). This helps us apply the product rule efficiently.

First, differentiate \( u(x) = x^2 \) to get \( u'(x) = 2x \). Then differentiate \( v(x) = (1 + x^2)^{1/2} \) using the chain rule, which when combined with our generalized power concept, gives \( v'(x) = \frac{x}{\sqrt{1+x^2}} \). By substituting back into the product rule formula, we get \( f'(x) = 2x(1+x^2)^{1/2} + x^2 \frac{x}{\sqrt{1+x^2}} \).

Finally, simplify the expression to find the derivative of the original function. This step often gives the solution the student is looking for.

The Chain Rule is pivotal in differentiating compositions of functions. Essentially, it says if you have a function \( g(h(x)) \), then the derivative is \( g'(h(x)) \cdot h'(x) \).

In our example, \( v(x) = (1+x^2)^{1/2} \) is a composition of the outer function \( g(x) = x^{1/2} \) and the inner function \( h(x) = 1+x^2 \). When differentiating \( v(x) \), the chain rule helps by first differentiating the outer function evaluated at the inner function, then multiplying by the derivative of the inner function:

• \( g'(h(x)) = \frac{1}{2} (1+x^2)^{-1/2} \)
• \( h'(x) = 2x \)

Therefore, applying the chain rule, the derivative \( v'(x) \) becomes \( \frac{1}{2}(1+x^2)^{-1/2} \cdot 2x = \frac{x}{\sqrt{1+x^2}} \). This neat transformation using the chain rule is crucial for differentiating complex nested functions within the broader calculus scope.