91Ó°ÊÓ

When differentiating functions that are products of two functions, the Product Rule is essential. The Product Rule states that if you have two functions, say \(u(x)\) and \(v(x)\), then the derivative of their product \(y = u(x) v(x)\) is given by:

• \[y' = u'(x)v(x) + u(x)v'(x)\]

To use the Product Rule, differentiate \(u(x)\) while keeping \(v(x)\) constant, then differentiate \(v(x)\) while keeping \(u(x)\) constant, and finally sum the two results. This is especially useful when dealing with complex functions broken down into simpler multiplicative components.

The Chain Rule is vital when differentiating composite functions. In essence, it helps us differentiate a function that is nested inside another function. If you have a composite function \(y = f(g(x))\), the Chain Rule states that its derivative is:

• \[y' = f'(g(x)) \times g'(x)\]

In simple terms, you differentiate the outer function and then multiply it by the derivative of the inner function. This rule often comes into play with functions that have exponents or when functions themselves are arguments to greater functions.

To apply the Chain Rule in the context of our example, differentiating \((x^2 + x + 1)^3\) involves first recognizing the 'outer' function is raising something to the power of 3, and the 'inner' function is \(x^2 + x + 1\). Subsequently, we differentiate the outer function and multiply it by the derivative of the inner function.

The Power Rule is one of the simplest yet powerful rules for differentiation. It deals with functions of the form \(x^n\), where \(n\) is any real number. According to the Power Rule, the derivative of \(x^n\) is:

• \[\frac{d}{dx}[x^n] = nx^{n-1}\]

This rule makes calculating the derivatives of polynomial functions straightforward by reducing the exponent by one and multiplying by the original exponent.
In our example, we use the Power Rule to differentiate \((x-1)^4\). Here, \(n\) is 4, so following the Power Rule, we multiply by the exponent and reduce the exponent by one.

Incorporating these rules, we apply the Product Rule first, then the Chain Rule to differentiate the first term \((x^2 + x + 1)^3\), and finally the Power Rule to differentiate the second term \((x-1)^4\). Combining these results according to the Product Rule gives us the complete derivative.