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To find the derivative of a composition of functions, we use the **chain rule**. The chain rule is a vital concept in differentiation. It allows us to differentiate composite functions, which are functions nested within each other. For instance, if we have a function like \( f(g(s)) \), we treat \( g(s) \) as an inner function and \( f(u) \) (where \( u = g(s) \)) as an outer function.
The chain rule states that the derivative of this composition is:

• Differentiate the outer function, keeping the inner function unchanged.
• Multiply this by the derivative of the inner function.

Here is the formula: \( (f(g(s)))' = f'(g(s)) \times g'(s) \)
In our problem, the outer function is \( (u)^{-2/3} \) and the inner function is \( s^4 + 3s^2 + 1 \). Differentiating such requires carefully applying the chain rule.

A fundamental rule in differentiation is the **power rule**. This rule is used to differentiate functions of the form \( x^n \), where \( n \) is any real number. The power rule states that: \( \frac{d}{dx} (x^n) = n \times x^{n-1} \)
This rule is extremely useful because it simplifies the process of finding derivatives of polynomial functions.
In our specific problem, the outer function is \( u^{-2/3} \) and by applying the power rule, it becomes:
\( \frac{d}{du} \big(u^{-2/3}\big) = -\frac{2}{3} u^{-5/3} \)
We also apply the power rule to the inner function \( s^4 + 3s^2 + 1 \) by differentiating each term separately:

• The derivative of \( s^4 \) is \( 4s^3 \)
• The derivative of \( 3s^2 \) is \( 6s \)
• The derivative of a constant \( 1 \) is 0.

Summing this up gives us the derivative of the inner function: \( \frac{d}{ds} \big(s^4 + 3s^2 + 1\big) = 4s^3 + 6s \). Understanding the power rule makes it easier to break down and differentiate complex polynomial functions.

Differentiation is the core concept that deals with finding the rate at which a function is changing at any given point. It is a fundamental tool in calculus. Differentiation is used to find the derivative of a function, which represents the slope or rate of change.
In our specific problem, we are asked to find the derivative of the function \( f(s) = \big( s^4 + 3s^2 + 1 \big)^{-2/3} \).
To do this, we follow a step-by-step approach:

• Identify the outer and inner functions.
• Apply the chain rule to differentiate the outer function while keeping the inner function intact.
• Differentiate the inner function separately.
• Combine both derivatives as per the chain rule formula.

Let's recap the steps for our function:

• Differentiate the outer function \( u^{-2/3} \) to get \( -\frac{2}{3} u^{-5/3} \).
• Differentiate the inner function \( s^4 + 3s^2 + 1 \) to get \( 4s^3 + 6s \).

Combine them to get the derivative:
\( f'(s) = -\frac{2}{3} \big( s^4 + 3s^2 + 1 \big)^{-5/3} \times (4s^3 + 6s) \).
Differentiation allows us to understand how functions change, which is crucial for solving many real-world problems.