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Question. If f is differentiable, find the limit Sol. Question. Find the limit: (1) (2) Sol. (1) (2) . The Chain Rule. Theorem If u=g(x) is differentiable at x=a and y=f(u) is

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  1. Question If f is differentiable, find the limit Sol

  2. Question • Find the limit: (1) (2) Sol. (1) (2)

  3. The Chain Rule Theorem If u=g(x) is differentiable at x=a and y=f(u) is differentiable at u=g(a). Then y=f(g(x)) is differentiable at x=a, and or,

  4. Derivative of power functions Ex. Differentiate Sol. Since can be written as the composition by the chain rule, we have

  5. Derivative of exponential functions Ex. Differentiate Sol. Since can be written into by the chain rule, we have

  6. Example Ex. Differentiate Sol. Let then By the chain rule, we have

  7. The power rule • The power rule combined with the chain rule • Ex. Find the derivative of • Sol.

  8. The chain rule If y=f(u), u=g(v) and v=h(x) are all differentiable, then y=f(g(h(x))) is differentiable and or,

  9. Example • Ex. Differentiate • Sol.

  10. Logarithmic differentiation Ex. Find the derivative of Sol. Not a power function, not an exponential function Since by product rule and chain rule, The method used here is called logarithmic differentiation

  11. Logarithmic differentiation In general, to differentiate we can take logarithm first to get then differentiating both sides Question: Find the derivative of Sol.

  12. Question Differentiate Sol.

  13. Implicit differentiation • Materials in textbook: page 227-233 • Outline • Derivative of implicit functions • Derivative of inverse trigonometric functions

  14. Expressions of functions • Explicit expression: y can be explicitly expressed in term of x, for example, • Implicit expression: x and y related by an equation, and can not solve y in terms of x explicitly, for example,

  15. Implicit differentiation Ex. Find if Sol. Differentiating both sides with respect to x, regarding y as a function of x, and using the chain rule, we get Solving the equation for we obtain Ex. Find an equation of the tangent line to the curve at the origin. Sol. is the slope

  16. Example Suppose y=f(x) is defined implicitly by (1) Find (2) Let find Sol. (1) (2)

  17. Homework 5 • Section 3.1: 45, 56, 57 • Section 3.2: 10, 21, 42 • Section 3.4: 11, 16, 38, 39, 42 • Section 3.5: 20, 28, 40 • Section 3.6: 10, 18

  18. Derivative of arcsine function • Ex. Find the derivative of Analysis. means differentiating will give Sol. Differentiating implicitly with respect to x, we obtain so

  19. Derivative of inverse functions • If x=f(y) is differentiable and then the inverse function is differentiable and or, Proof.

  20. Example Similarly,

  21. Higher derivatives • The derivative of is called the second derivative of f and denoted by or • Recursively, we can define the third derivative and generally the nth derivative • Interpretation: for example, if s(t) is displacement, then is velocity, is acceleration and is jerk.

  22. Example If then

  23. Example Find if Sol. At x=0, y=1, and thus

  24. Example If find Sol.

  25. Example Find if Sol. Using the trigonometry identity Suppose then Therefore

  26. Question Find if Hint: Sol.

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