Product Rule for three functi​​ons Assume f, g, and h are differentiable at x.
a. Use the Product Rule (twice) to find a formula for d/dx (f(x)g(x)h(x)).

Verified step by step guidance

1. Step 1: Recall the Product Rule for two functions, which states that if u(x) and v(x) are differentiable functions, then the derivative of their product is given by \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \).
2. Step 2: To find the derivative of the product of three functions \( f(x)g(x)h(x) \), first consider \( u(x) = f(x) \) and \( v(x) = g(x)h(x) \). Apply the Product Rule to these two functions.
3. Step 3: Differentiate \( v(x) = g(x)h(x) \) using the Product Rule again. Let \( u(x) = g(x) \) and \( v(x) = h(x) \), so \( \frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x) \).
4. Step 4: Substitute the result from Step 3 into the expression obtained in Step 2. This gives \( \frac{d}{dx}[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)(g'(x)h(x) + g(x)h'(x)) \).
5. Step 5: Simplify the expression from Step 4 to obtain the final formula: \( \frac{d}{dx}[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) \).