Given that f(1) = 5, f′(1) = 4, g(1) = 2, and g′(1) = 3 , find d/dx (f(x)g(x))∣ ∣x=1 and d/dx (f(x) / g(x)) ∣ x=1.

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Step 1: To find the derivative of the product of two functions, use the product rule. The product rule states that if you have two functions f(x) and g(x), then the derivative of their product is given by: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x).

Step 2: Apply the product rule to the given functions at x = 1. Substitute f(1) = 5, f'(1) = 4, g(1) = 2, and g'(1) = 3 into the product rule formula: (f(x)g(x))'|_{x=1} = f'(1)g(1) + f(1)g'(1).

Step 3: To find the derivative of the quotient of two functions, use the quotient rule. The quotient rule states that if you have two functions f(x) and g(x), then the derivative of their quotient is given by: (f(x)/g(x))' = (f'(x)g(x) - f(x)g'(x)) / (g(x))^2.

Step 4: Apply the quotient rule to the given functions at x = 1. Substitute f(1) = 5, f'(1) = 4, g(1) = 2, and g'(1) = 3 into the quotient rule formula: (f(x)/g(x))'|_{x=1} = (f'(1)g(1) - f(1)g'(1)) / (g(1))^2.

Step 5: Simplify the expressions obtained from the product and quotient rules to find the derivatives at x = 1.

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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 the product of two functions. It states that if you have two functions, f(x) and g(x), the derivative of their product is given by f'(x)g(x) + f(x)g'(x). This rule is essential for solving the first part of the question, where we need to find the derivative of the product f(x)g(x) at x=1.

Quotient Rule

The Quotient Rule is another important differentiation rule that applies when dividing two functions. It states that if you have a function h(x) = f(x)/g(x), the derivative is given by (f'(x)g(x) - f(x)g'(x)) / (g(x))^2. This rule is crucial for solving the second part of the question, where we need to find the derivative of the quotient f(x)/g(x) at x=1.

Evaluation of Derivatives

Evaluating derivatives at a specific point involves substituting the value of x into the derivative expression after applying the appropriate differentiation rules. In this question, we will substitute x=1 into the results obtained from the Product and Quotient Rules to find the specific values of the derivatives at that point, which is necessary for completing the problem.