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When finding the derivative of composite functions like \( y = \sqrt{f(x)} \), understanding the concept behind the Chain Rule is essential. The Chain Rule helps us differentiate functions where one function is inside another. Suppose we have two functions, \( f \) and \( g \), combined to form \( h(x) = f(g(x)) \). The Chain Rule states that the derivative of \( h \) with respect to \( x \), noted as \( h'(x) \), is:

• \( h'(x) = f'(g(x)) \cdot g'(x) \)

In simpler terms, you first differentiate the outer function \( f \) as if the inner function \( g(x) \) is a constant, then multiply by the derivative of the inner function \( g \).
In our example, \( y = \sqrt{f(x)} \), we can think of this as \( y = f(x)^{1/2} \). Using the Chain Rule involves:

• Differentiating the outer function \( (x^{1/2}) \), resulting in \( 1/2 \cdot x^{-1/2} \).
• Multiplying by the derivative of the inner function, \( f'(x) \).

This process allows us to systematically break down and find the derivative of more complex composite functions.

Differentiation is all about understanding how a function's value changes. The Power Rule is a fundamental technique for finding these changes when a function is expressed as a power of \( x \). The rule is quite simple: if \( y = x^n \), then \( \frac{dy}{dx} = nx^{n-1} \).
The Power Rule simplifies taking derivatives because it provides a straightforward formula to apply. Let's examine how it works using our exercise:

• We rewrote the function \( y = \sqrt{f(x)} \) as \( y = f(x)^{1/2} \).
• Applying the Power Rule to the outer part, \( f(x)^{1/2} \), gives us \( 1/2 \cdot f(x)^{-1/2} \).
  The exponent is reduced by 1, turning \( 1/2 \) into \( -1/2 \).

This application of the Power Rule is crucial when we are dealing with any functions that involve powers, making the process of differentiation much quicker and more precise.

For any function to be differentiable, it must be smooth and continuous without any sharp points or breaks. Differentiability ensures we can calculate a derivative for most points on the function.
The concept of differentiability is essential because:

• It informs us about the behavior of the function and whether it can be effectively analyzed using calculus.
• A differentiable function is also continuous, but not all continuous functions are differentiable.

In our example, the function \( f(x) \) is stated to be differentiable. This means we can apply calculus rules like the Chain Rule and Power Rule without encountering undefined operations or errors. Differentiability guarantees that wherever we pick a point \( x \), we are able to compute a meaningful derivative, thereby predicting accurately how the function \( y = \sqrt{f(x)} \) changes with small changes in \( x \). This is critical when solving real-world problems where smooth and predictable behavior of a system or phenomenon is expected.