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The chain rule is a fundamental concept in calculus differentiation. It's used to find the derivative of composite functions, where one function is nested inside another. To apply the chain rule, you need to differentiate both the outer and the inner functions separately. The mathematical expression for this is

• \( \frac{dy}{dx} = \frac{df}{du} \times \frac{du}{dx} \)

In this expression:

• \( f(u) \) is the outer function
• \( g(x) \) is the inner function

By multiplying the derivative of the outer function by the derivative of the inner function, you effectively compute the derivative of the original composite function. Imagine peeling an onion: you handle the outer layer first before getting to the inner layers, reflecting how the chain rule tackles differentiation step-by-step.

Understanding function composition is crucial as it helps to identify which functions are nested within each other. When you see an expression like \( y = (2x + 5)^{-1/2} \), it can be broken down into simpler parts, or components. This breaking down is called 'function composition'.

• The outer function, in this case, is \( f(u) = u^{-1/2} \)
• The inner function is \( g(x) = 2x + 5 \)

Function composition means substituting one function into another. Here, \( f(u) \) operates on \( g(x) \), creating a compound expression: \( y = f(g(x)) \). Recognizing these compositions allows you to apply the chain rule effectively. This ability to simplify a problem by seeing its building blocks is key in calculus differentiation.

The computation of derivatives involves calculating the rate at which a function changes with respect to its variable. In our example, we are tasked with finding \( \frac{dy}{dx} \), the derivative of \( y \) with respect to \( x \). To do this:
First, differentiate the outer function \( f(u) = u^{-1/2} \) by applying the power rule. This yields \( f'(u) = -1/2 \cdot u^{-3/2} \).
Next, differentiate the inner function \( g(x) = 2x + 5 \), which is straightforward: \( g'(x) = 2 \).
Finally, combine these using the chain rule:

• Multiply \( f'(u) \) by \( g'(x) \), giving \(-1/2 \cdot (2x + 5)^{-3/2} \cdot 2 \).

This process of derivative computation shows how complex functions can be managed by breaking them into simpler, understandable parts and applying basic differentiation rules.