Textbook Question
Derivatives from graphs Use the figure to find the following derivatives. <IMAGE>
d/dx (xg(x)) | x=2
3. Techniques of Differentiation
Product and Quotient Rules
9:03 minutesProblem 99a
Textbook Question
Product Rule for three functions 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 guidance1
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 \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \).
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.
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 \( \frac{d}{dx}[g(x)h(x)] = g'(x)h(x) + g(x)h'(x) \).
Step 4: Substitute the result from Step 3 into the expression obtained in Step 2. This gives \( \frac{d}{dx}[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)(g'(x)h(x) + g(x)h'(x)) \).
Step 5: Simplify the expression from Step 4 to obtain the final formula: \( \frac{d}{dx}[f(x)g(x)h(x)] = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product Rule
The Product Rule is a fundamental principle in calculus used to differentiate products of functions. It states that if you have two differentiable functions, f(x) and g(x), the derivative of their product is given by d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x). This rule can be extended to more than two functions, allowing for the differentiation of products involving three or more functions.
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Differentiability
A function is said to be differentiable at a point if it has a defined derivative at that point, meaning it has a tangent line that is not vertical. Differentiability implies continuity, but not vice versa. For the Product Rule to apply, all functions involved must be differentiable at the point of interest, ensuring that their derivatives can be computed and combined appropriately.
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Chain of Derivatives
When applying the Product Rule to multiple functions, it is essential to understand how to manage the derivatives of each function in the product. For three functions, f(x), g(x), and h(x), the derivative is found by applying the Product Rule iteratively. This involves taking the derivative of one function while keeping the others constant, and then summing the results, which requires careful organization of terms to ensure all combinations are accounted for.
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