91Ó°ÊÓ

In calculus, the power rule is a straightforward and fundamental tool for differentiating functions. It applies to functions of the form \(x^n\) where \(n\) is any real number. The power rule states that the derivative of \(x^n\) is \(nx^{n-1}\). This means you multiply by the exponent \(n\) and then decrease the exponent by one. This rule simplifies finding derivatives because it turns potentially complex expressions into manageable forms.
To apply the power rule effectively, identify terms that follow the \(x^n\) structure. For example, in the original function \(u(x)=(6.4x-1)^2\), we see an exponent of 2. Using the power rule:

• The derivative \(u'(x)\) becomes \(2(6.4x-1)^{1}\times 6.4\).
• Essentially, multiply the original exponent by the inner function's derivative.

This quick conversion is particularly useful for polynomial expressions encountered in calculus.

The chain rule is an essential derivative technique used when functions are nested within each other. Essentially, it allows you to differentiate composite functions systematically. The chain rule states that the derivative of a composite function \(f(g(x))\) is \(f'(g(x)) \cdot g'(x)\). Think of it as peeling layers of an onion, removing one layer at a time.
For the given function \(f(x)=\left[(6.4x-1)^{2}+(5.4x-2)^{3}\right]^{2}\), the chain rule comes into play because we have a function raised to a power, and those bases themselves are also composite functions. The steps:

• Differentiate the outer function, leaving the inner functions intact initially.
• Multiply by the derivative of the inner functions, which involves the power rule in our case.

So, the derivative of the outer function is \(2(u(x) + v(x))^{1}\), which is then multiplied by the inner derivatives \(u'(x)+v'(x)\), as shown in the step-by-step solution. This approach efficiently tackles complex nested functions.

Understanding differentiation techniques is crucial for solving calculus problems accurately and efficiently. Differentiation refers to the process of finding a function's derivative, which represents the rate of change of the function. In this exercise, a combination of techniques is involved:

• **Power Rule**: Used for polynomial-like parts, as seen in finding \(u'(x)\) and \(v'(x)\).
• **Chain Rule**: Used to differentiate composite functions, especially when layers of functions are nested.

By employing these techniques together, we can tackle more complicated expressions. Start by breaking down the function into its components and identifying which rules apply to each part. Ensure each layer is differentiated correctly using corresponding rules.
A systematic approach enhances understanding and efficiency, helping you apply the right steps in sequence for accurate results.