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Mathematics LibreTexts

The Product Rule

Consider the product of two simple functions, say \( f(x)=(x^2+1)(x^3-3x)\). An obvious guess for the derivative of \(f\) is the product of the derivatives of the constituent functions: \( (2x)(3x^2-3)=6x^3-6x\).

Is this correct? We can easily check, by rewriting \(f\) and doing the calculation in a way that is known to work. First, \( f(x)=x^5-3x^3+x^3-3x=x^5-2x^3-3x\), and then \( f'(x)=5x^4-6x^2-3\). Not even close! What went "wrong''? Well, nothing really, except the guess was wrong.

So the derivative of \(f(x)g(x)\) is NOT as simple as \(f'(x)g'(x)\). Surely there is some rule for such a situation? There is, and it is instructive to "discover'' it by trying to do the general calculation even without knowing the answer in advance.

\[\eqalign{ {d\over dx}(&f(x)g(x)) = \lim_{\Delta x \to0} {f(x+\Delta x)g(x+\Delta x) - f(x)g(x)\over \Delta x}\cr& =\lim_{\Delta x \to0} {f(x+\Delta x)g(x+\Delta x)-f(x+\Delta x)g(x) + f(x+\Delta x)g(x)- f(x)g(x)\over \Delta x}\cr & =\lim_{\Delta x \to0} {f(x+\Delta x)g(x+\Delta x)-f(x+\Delta x)g(x)\over \Delta x} + \lim_{\Delta x \to0} {f(x+\Delta x)g(x)- f(x)g(x)\over \Delta x}\cr & =\lim_{\Delta x \to0} f(x+\Delta x){ g(x+\Delta x)-g(x)\over \Delta x} + \lim_{\Delta x \to0} {f(x+\Delta x)- f(x)\over \Delta x}g(x)\cr & =f(x)g'(x) + f'(x)g(x)\cr }\]

A couple of items here need discussion. First, we used a standard trick, "add and subtract the same thing'', to transform what we had into a more useful form. After some rewriting, we realize that we have two limits that produce \(f'(x)\) and \(g'(x)\). Of course, \(f'(x)\) and \(g'(x)\) must actually exist for this to make sense. We also replaced \( \lim_{\Delta x\to0}f(x+\Delta x)\) with \(f(x)\)---why is this justified?

What we really need to know here is that \( \lim_{\Delta x\to 0}f(x+\Delta x)=f(x)\), or in the language of section 2.5, that \(f\) is continuous at \(x\). We already know that \(f'(x)\) exists (or the whole approach, writing the derivative of \(fg\) in terms of \(f'\) and \(g'\), doesn't make sense). This turns out to imply that \(f\) is continuous as well. Here's why:

\[ \eqalign{ \lim_{\Delta x\to 0} f(x+\Delta x) &= \lim_{\Delta x\to 0} (f(x+\Delta x) -f(x) + f(x))\cr& = \lim_{\Delta x\to 0} {f(x+\Delta x) -f(x)\over \Delta x}\Delta x + \lim_{\Delta x\to 0} f(x)\cr& =f'(x)\cdot 0 + f(x) = f(x)\cr }\]

 

To summarize: the product rule says that

\[{d\over dx}(f(x)g(x)) = f(x)g'(x) + f'(x)g(x). \]

Returning to the example we started with, let

\[ f(x)=(x^2+1)(x^3-3x).\]

Then

\[ f'(x)=(x^2+1)(3x^2-3)+(2x)(x^3-3x)=3x^4-3x^2+3x^2-3+2x^4-6x^2= 5x^4-6x^2-3,\]

as before. In this case it is probably simpler to multiply \(f(x)\) out first, then compute the derivative; here's an example for which we really need the product rule.

Example 3.3.1
Compute the derivative of \( f(x)=x^2\sqrt{625-x^2}\).
SOLUTION
We have already computed
\[ {d\over dx}\sqrt{625-x^2}={-x\over\sqrt{625-x^2}}.\]
Now
\[f'(x)=x^2{-x\over\sqrt{625-x^2}}+2x\sqrt{625-x^2}= {-x^3+2x(625-x^2)\over \sqrt{625-x^2}}= {-3x^3+1250x\over \sqrt{625-x^2}}. \]

Exercises 3.3

In 1--4, find the derivatives of the functions using the product rule.

Ex 3.3.1 \( x^3(x^3-5x+10)\) (answer)

Ex 3.3.2 \( (x^2+5x-3)(x^5-6x^3+3x^2-7x+1)\) (answer)

Ex 3.3.3 \( \sqrt{x}\sqrt{625-x^2}\) (answer)

Ex 3.3.4 \(\displaystyle {\sqrt{625-x^2}\over x^{20}}\) (answer)

Ex 3.3.5 Use the product rule to compute the derivative of \( f(x)=(2x-3)^2\). Sketch the function. Find an equation of the tangent line to the curve at \(x=2\). Sketch the tangent line at \(x=2\). (answer)

Ex 3.3.6 Suppose that \(f\), \(g\), and \(h\) are differentiable functions. Show that \((fgh)'(x) = f'(x) g(x)h(x) + f(x)g'(x) h(x) + f(x) g(x) h'(x)\).

Ex 3.3.7 State and prove a rule to compute \((fghi)'(x)\), similar to the rule in the previous problem.

Remark 3.3.2 {Product notation} Suppose \( f_1 , f_2 , \ldots f_n\) are functions. The product of all these functions can be written \( \prod _{k=1 } ^n f_k.\) This is similar to the use of \( \sum\) to denote a sum. For example,

\[\prod _{k=1 } ^5 f_k =f_1 f_2 f_3 f_4 f_5\]

and

\[ \prod _ {k=1 } ^n k = 1\cdot 2 \cdot \ldots \cdot n = n!.\]

We sometimes use somewhat more complicated conditions; for example

\[\prod _{k=1 , k\neq j } ^n f_k\]

denotes the product of \( f_1\) through \( f_n\) except for \( f_j\). For example,

\[\prod _{k=1 , k\neq 4} ^5 x^k = x\cdot x^2 \cdot x^3 \cdot x^5 = x^{11}.\]

Ex 3.3.8 The generalized product rule says that if \( f_1 , f_2 ,\ldots ,f_n\) are differentiable functions at \(x\) then

\[{d\over dx}\prod _{k=1 } ^n f_k(x) = \sum _{j=1 } ^n \left(f'_j (x) \prod _{k=1 , k\neq j} ^n f_k (x)\right).\]

Verify that this is the same as your answer to the previous problem when \(n=4\), and write out what this says when \(n=5\).