微积分第三章课件 DIFFERENTIATION RULES.ppt

Solution (a) Differentiating both sides of the equation with respect to x gives that (x2+y2)’=(25)’, 2x+2y·y’=0, and y’=-x/y, y≠0. (b) At (3, 4), we find that y’=-3/4. The equation of the tangent to the circle at (3, 4) is y-4 = (-3/4)(x-3) or 3x+4y =25 Example 2 Solution Differentiating implicitly with respect to x and remembering that y is a function of x, we have Derivatives of Inverse Trigonometric Functions Recall the definition of the arcsine function: Differentiating siny = x implicitly with respect to x, we get Differentiating tany = x implicitly with respect to x, we have Example 3 The derivatives of inverse trigonometric functions are given in the following table. 3.7 Higher Derivatives If f is a differentiable function, then its derivative f’ is also a function, so f’ may have a derivative of its own, denoted by (f’)’=f’’. The new function f’’ is called the second derivative of f. Using Leibniz notation, we write the second derivative of y = f(x) as Example 1 The third derivative f’’’ is the derivative of the second derivative: (f’’)’=f’’’. Using Leibniz notation, we write the third derivative of y = f(x) as In general, the nth derivative of f is denoted by f(n) and is obtained from f by differentiating n times. Example 2 Example 3 Solution The following example shows how to find the second derivative of a function that is defined implicitly. Example 4 Solution Differentiating the equation implicitly with respect to x, we have * CHAPTER 3 DIFFERENTIATION RULES 3.1 Derivatives of Polynomials and Exponential Functions 3.2 The Product and Quotient Rules 3.3 Rates of Change in the Natural and Social Sciences 3.4 Derivatives of Trigonometric Functions 3.5 The Chain Rule 3.6 Implicit Differentiation 3.7 Higher Derivatives 3.8 derivatives of Logarithmic Functions 3.9 Hyperbolic Functions 3.10 Related Rates 3.11 Linear Approximations and Differentials 3.1 Derivatives of