Calc03 8 (1)
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The document discusses derivatives of inverse trigonometric functions. It provides the formula that the derivative of the inverse function evaluated at x is equal to the reciprocal of the derivative of the original function evaluated at y. As an example, it shows that the derivative of arcsin(x) evaluated at 2 is equal to the reciprocal of the derivative of sin(y) evaluated at the value where sin(y)=2. The document also mentions using implicit differentiation to find derivatives of inverse trigonometric functions.
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This document discusses different types of proportional relationships between two or more variables, including direct, inverse, and multiple proportionality. It provides examples of directly and inversely proportional functions, explaining that a directly proportional function results in a constant ratio between variables in each row, while an inversely proportional function results in a constant product. The document also introduces the concept of compound proportionality between three or more variables and shows how to solve problems using direct and inverse rules of three.
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Calc03 8 (1)
- 1. 3.8 Derivatives of Inverse Trig Functions Lewis and Clark Caverns, Montana
- 2. We can find the inverse function as follows: Switch x and y . At x = 2 : To find the derivative of the inverse function:
- 3. At x = 2 : At x = 4 : Slopes are reciprocals.
- 4. Because x and y are reversed to find the reciprocal function, the following pattern always holds: is equal to the reciprocal of Slopes are reciprocals. Derivative Formula for Inverses: evaluated at the derivative of evaluated at . The derivative of
- 5. A typical problem using this formula might look like this: Given: Find: Derivative Formula for Inverses:
- 6. We can use implicit differentiation to find:
- 7. We can use implicit differentiation to find: But so is positive.
- 8. We could use the same technique to find and . 1 sec d x dx
- 9. Your calculator contains all six inverse trig functions. However it is occasionally still useful to know the following: