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Math 180

Explore proofs and derivations of derivatives with various functions in calculus, including trigonometric, exponential, logarithmic, and hyperbolic functions. Discover how to find derivatives using limits and geometric arguments.

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Math 180

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  1. Math 180 Packet #10 Proving Derivatives

  2. Proof of Power Rule Suppose . Then,

  3. Proof of Power Rule Suppose . Then,

  4. Proof of Power Rule Suppose . Then,

  5. Proof of Power Rule Suppose . Then,

  6. Proof of Power Rule Suppose . Then,

  7. Proof of Power Rule Suppose . Then,

  8. Proof of Power Rule Suppose . Then,

  9. Trigonometric Functions To figure out the derivatives of and by the definition, we need these two limits: and These two limits can be proven using a geometric argument. Then we can find the derivatives of and as follows:

  10. Trigonometric Functions To figure out the derivatives of and by the definition, we need these two limits: and These two limits can be proven using a geometric argument. Then we can find the derivatives of and as follows:

  11. Trigonometric Functions

  12. Trigonometric Functions

  13. Trigonometric Functions

  14. Trigonometric Functions

  15. Trigonometric Functions

  16. Trigonometric Functions

  17. Trigonometric Functions

  18. Trigonometric Functions

  19. Trigonometric Functions

  20. Trigonometric Functions

  21. Trigonometric Functions But what about and the rest of the gang? Ex 1.Prove that the derivative of is by using the derivatives of and .

  22. Exponential Functions To find the derivative of by the definition, we need the following limit: This limit is sometimes taken to just be definitional. That is, is the number where .

  23. Exponential Functions To find the derivative of by the definition, we need the following limit: This limit is sometimes taken to just be definitional. That is, is the number where .

  24. Exponential Functions Then,

  25. Exponential Functions Then,

  26. Exponential Functions Then,

  27. Exponential Functions Then,

  28. Exponential Functions Then,

  29. Exponential Functions Here’s how to get the derivative of :

  30. Exponential Functions Here’s how to get the derivative of :

  31. Exponential Functions Here’s how to get the derivative of :

  32. Exponential Functions Here’s how to get the derivative of :

  33. Exponential Functions Here’s how to get the derivative of :

  34. Logarithmic Functions To find the derivative of logarithms, we can use implicit differentiation. Ex 2.Prove that the derivative of is .

  35. Logarithmic Functions Note: We can extend our results to include negative -values:

  36. Logarithmic Functions Here’s how to get the derivative of .

  37. Logarithmic Functions Here’s how to get the derivative of .

  38. Logarithmic Functions Here’s how to get the derivative of .

  39. Logarithmic Functions Here’s how to get the derivative of .

  40. Logarithmic Functions Here’s how to get the derivative of .

  41. Inverse Trigonometric Functions We can use implicit differentiation with inverse trig functions, too. Ex 3.Prove that the derivative of is .

  42. Hyperbolic Functions Recall that and . Ex 4.Prove that the derivative of is .

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