Final answer:
The derivative of y=(4x^2 −8x+3)^4 is found using the chain rule. The equation of the tangent line at the point (2,81) is y = x, indicating the slope is 1 and it passes through the point (2,81).
Step-by-step explanation:
The student asked to find the derivative of the function y=(4x2 −8x+3)4 using the chain rule (extended power rule) and to provide the derivative in a specified form. Applying the chain rule, the derivative can be found by taking the derivative of the outer function and multiplying it by the derivative of the inner function. The resulting derivative y' will have the form y' = A(Bx2 −Cx+D)E (Fx−G).
To find the equation of the tangent line at the point (2,81), we must evaluate the derivative at x = 2 to get the slope of the tangent. Then, we use the point-slope form of a line y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point through which the line passes.
It's notable that in the scenario provided, the equation for the tangent line results in y = x, indicating that the slope of the tangent line at the point (2,81) will be 1 and it passes through that point.