Answer
\[
\frac{d}{dx} \big(u(x)v(x)\big)
= v(x)\frac{du}{dx} + u(x)\frac{dv}{dx},
\]
Work Step by Step
\[
f(a,b) = ab.
\]
Let both variables depend on a single variable \(x\), where
\[
a = u(x), \qquad b = v(x).
\]
Then the composition becomes
\[
f(u(x), v(x)) = u(x)v(x).
\]
By the chain rule for functions of two variables,
\[
\frac{d}{dx} f(a,b) = f_a(a,b)\frac{da}{dx} + f_b(a,b)\frac{db}{dx},
\]
where \(f_a\) and \(f_b\) denote partial derivatives.
Since \(f(a,b) = ab\), we have
\[
f_a(a,b) = b, \qquad f_b(a,b) = a.
\]
Substituting,
\[
\frac{d}{dx} \big(u(x)v(x)\big)
= v(x)\frac{du}{dx} + u(x)\frac{dv}{dx},
\]
which is exactly the product rule.
