Definition of Derivative:

Rule: Assume f(x) and g(x) are two differentiable functions, the basic rules of derivative are

Definition of Critical Point: is called a critical point of f(x), if

If on some interval I, then f(x) is increasing on the interval I. Similarly, if on some interval I, then f(x) is decreasing on the interval I.

Tangent Line: Assume f(x) is a differentiable function on the interval I, then the tangent line of f(x) at the point is where is the slope of the tangent line.

Derivative of Parameter Functions: Assume y=y(t) and x=x(t), the derivative is because the Chain Rule of derivatives.

Question 1. Calculate the tangent line of the curve at the point (16,16).

Method (i). Take the derivative of the equation at the both sides, we get

Assume x=y=16, we have the derivative That means the tangent line of the curve at the point (16,16) is y-16=-(x-16). i.e. y=-x+32.

Method (ii). From the equation, we know , then calculating the derivative directly. i.e.

Therefore,

Method (iii). Making the substitution then (16,16) corresponds to From the derivative of the parameter functions, we know

If we assume then

Method (iv). Geometric Intuition. Since the equation is a symmetric graph with the line y=x, and (16,16) is also on the symmetric line. Therefore, the slope of the curve at the point (16,16) is -1. Hence, the tangent line is y=-x+32.

Question 2. Let and Find dy/dx and express your answer in terms of

Method (i). ,

Method (ii).