Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.
y = f(x)g(x)

Verified step by step guidance

1. Step 1: Identify the given information. We have two functions, f(x) and g(x), with their respective tangent lines at x=2. The tangent line to f(x) at x=2 is y=4x+1, which implies that f'(2)=4 and f(2) is the y-value when x=2, which can be found by substituting x=2 into the equation of the tangent line. Similarly, the tangent line to g(x) at x=2 is y=3x−2, which implies that g'(2)=3 and g(2) is the y-value when x=2, found by substituting x=2 into the equation of the tangent line.
2. Step 2: Use the product rule to find the derivative of y=f(x)g(x). The product rule states that if y=u(x)v(x), then y'=u'(x)v(x) + u(x)v'(x). Here, u(x)=f(x) and v(x)=g(x), so y'=f'(x)g(x) + f(x)g'(x).
3. Step 3: Substitute x=2 into the derivative found in Step 2. This gives y'(2)=f'(2)g(2) + f(2)g'(2).
4. Step 4: Substitute the known values from Step 1 into the expression from Step 3. We have f'(2)=4, g'(2)=3, and the values of f(2) and g(2) can be found from the tangent line equations.
5. Step 5: Use the result from Step 4 to write the equation of the tangent line to y=f(x)g(x) at x=2. The equation of a tangent line is y-y_1=m(x-x_1), where m is the slope found in Step 4, and (x_1, y_1) is the point on the curve at x=2, which can be found by evaluating y=f(x)g(x) at x=2.