Products with Composite Factors
A product can contain factors that are more complicated than a single power of \(x\). For example, in \((x^2+1)^3(2x-1)^4\), each factor is a power of an expression involving \(x\). Differentiating the whole expression requires attention to two different structures: the factors are multiplied, and each factor is a composite function.
The method is to differentiate each factor using the Chain Rule, then combine those derivatives using the Product Rule. Keeping those steps distinct helps prevent two common errors: omitting an inner derivative and forgetting one of the terms from the product rule.
The Rules and Their Conditions
The Product Rule applies to functions \(F\) and \(G\) that are differentiable at the point under consideration. It states that the derivative of their product has two terms: differentiate the first factor and leave the second unchanged, then leave the first unchanged and differentiate the second.
The Chain Rule applies to a composition \(f(g(x))\): if \(g\) is differentiable at \(x\) and \(f\) is differentiable at \(g(x)\), then the derivative is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
In particular, for a positive integer \(m\), the power rule together with the Chain Rule gives
where \(u\) is differentiable. The factor \(u'(x)\) is essential: it accounts for the rate of change of the expression inside the power.
A Formula for Products of Powered Factors
Suppose \(u\) and \(v\) are differentiable functions and \(m,n\) are positive integers. Set \(F(x)=u(x)^m\) and \(G(x)=v(x)^n\). The Product Rule gives the derivative of \(FG\), while the Chain Rule gives the derivatives of \(F\) and \(G\). Substituting those derivatives produces a useful formula.
Here is the derivation in full. By the Product Rule, differentiate the first factor while keeping the second fixed, then add the first factor times the derivative of the second. The Chain Rule evaluates those two factor derivatives as shown below.
This derivation explains why both kinds of differentiation are needed. The Product Rule accounts for the two terms, one for each factor. Within each term, the Chain Rule differentiates the relevant composite factor.
The displayed formula is stated for positive integer exponents. If an exponent is zero, its powered factor is the constant function \(1\), so treat that factor as \(1\) and differentiate any remaining factor separately. This avoids using a right-hand-side power such as \(u(x)^{-1}\) in a case where it might be undefined.
Worked Example: Two Powered Composite Factors
Worked Example: Differentiate \((x^2+1)^3(2x-1)^4\)
Let \(F(x)=(x^2+1)^3\) and \(G(x)=(2x-1)^4\). The original function is \(F(x)G(x)\), so first find each factor’s derivative and then apply the Product Rule.
For the first factor, the outer power has exponent \(3\), and the inner function is \(x^2+1\), whose derivative is \(2x\). Therefore,
For the second factor, the outer power has exponent \(4\), and the inner function is \(2x-1\), whose derivative is \(2\). Thus,
Now use the Product Rule, retaining the undifferentiated factor in each term:
Both terms contain \(2(x^2+1)^2(2x-1)^3\). Factoring it out gives
To check the result at \(x=1\), the original factors have values \(F(1)=2^3=8\) and \(G(1)=1^4=1\). Their derivatives have values \(F'(1)=6(1)(2)^2=24\) and \(G'(1)=8(1)^3=8\). The Product Rule therefore gives \(24(1)+8(8)=88\). The factored derivative gives \(2(2)^2(1)^3(10-3+4)=8(11)=88\), as required.
Worked Example: Different Inner Derivatives
Worked Example: Differentiate \((3x-2)^2(x^2+4)^3\)
There are two factors. The inner function in the first is \(3x-2\), with derivative \(3\); the inner function in the second is \(x^2+4\), with derivative \(2x\). Apply the Chain Rule to each factor:
Apply the Product Rule, placing each factor derivative in turn while retaining the other factor:
For a check at \(x=0\), the factors are \((-2)^2=4\) and \(4^3=64\). Their derivatives are \(6(-2)=-12\) and \(6(0)(4)^2=0\). The Product Rule gives \((-12)(64)+(4)(0)=-768\). Substitution into the derivative expression gives \(6(-2)(4)^3+0=-768\), confirming the calculation.
Worked Example: Checking a Product at a Point
Worked Example: Differentiate and evaluate at \(x=1\)
Differentiate \(H(x)=(x^2+2x)^5(4x-3)^2\), then find \(H'(1)\). Write the factors as \(F(x)=(x^2+2x)^5\) and \(G(x)=(4x-3)^2\). The inner derivatives are \(2x+2\) and \(4\), respectively. Hence
Use the Product Rule and substitute these two derivatives:
At \(x=1\), \(x^2+2x=1+2=3\), \(2x+2=4\), and \(4x-3=1\). The first term is \(5(3)^4(4)(1)^2=5(81)(4)=1620\). The second term is \(8(3)^5(1)=8(243)=1944\). Therefore, \(H'(1)=1620+1944=3564\).
Check using the factor values and derivatives: \(F(1)=3^5=243\), \(G(1)=1^2=1\), \(F'(1)=5(3)^4(4)=1620\), and \(G'(1)=8(1)=8\). Thus \(F'(1)G(1)+F(1)G'(1)=1620(1)+243(8)=1620+1944=3564\), matching the result.
A Common Mistake: Leaving Out an Inner Derivative
Consider the first factor in the opening example. A common incorrect calculation is
This differentiates the outer power but ignores the inner function \(x^2+1\). The Chain Rule requires multiplication by the inner derivative, which is \(2x\). The correct calculation is
Another error is to write only \(F'G\) when differentiating \(FG\). That omits the change in \(G\). The full Product Rule has both \(F'G\) and \(FG'\); each term must retain the factor that was not differentiated.
A Reliable Procedure
For a product of two composite factors, organize the calculation in the following order. This makes it easier to see which rule is being used at each stage.
Write the expression as \(F(x)G(x)\), keeping each composite factor intact.
Use the Chain Rule, including the derivative of the inner expression in each factor.
Form \(F'G+FG'\); do not omit either term or the undifferentiated factor.
Combine like factors only after the product-rule terms are correct. When evaluating at a point, check the factor values and derivative values separately.
Why the Distinction Matters
The Product Rule and the Chain Rule describe different operations. The Product Rule concerns two functions multiplied together. The Chain Rule concerns one function applied to another. In \((x^2+1)^3(2x-1)^4\), the two powered expressions are multiplied, so the whole expression needs the Product Rule. Each powered expression is also a composition, so differentiating either one needs the Chain Rule.
Recognizing both structures prevents the tempting but incorrect idea that a product can be differentiated by treating it as one large power or by differentiating only one of its factors. A correct derivative records every way the product can change: the first factor can change while the second remains, and the second can change while the first remains.
Check Your Understanding
For each question, identify the relevant composite factors and write enough steps to show how the rules apply.
- Differentiate \((x^2+3)^2(5x+1)^3\), showing the derivative of each factor before applying the Product Rule.
- In differentiating \((2x-1)^4(x^2+1)^2\), what inner derivative is required for each powered factor?
- Explain why the derivative of \(F(x)G(x)\) has two terms rather than just \(F'(x)G(x)\).
- For \(H(x)=(x+2)^3(x^2+1)^2\), calculate \(H'(0)\) by finding the factor values and factor derivatives at zero.
- If one factor is raised to the zero power and is defined as the constant \(1\), how should it be handled when differentiating the product?
