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.

Verified step by step guidance

1. 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).
2. 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).
3. 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.
4. 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.
5. Step 5: Simplify the expressions obtained from the product and quotient rules to find the derivatives at x = 1.