3 8 Derivatives of Inverse Trig Functions At
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3. 8 Derivatives of Inverse Trig Functions
At x = 2: We can find the inverse function as follows: Switch x and y. To find the derivative of the inverse function:
Slopes are reciprocals. At x = 2: At x = 4:
Slopes are reciprocals. Because x and y are reversed to find the reciprocal function, the following pattern always holds: Derivative Formula for Inverses: The derivative of evaluated at is equal to the reciprocal of the derivative of evaluated at .
A typical problem using this formula might look like this: Given: Find: Derivative Formula for Inverses:
A function has an inverse only if it is one-to-one. We remember that the graph of a one-to-one function passes the horizontal line test as well as the vertical line test. We notice that if a graph fails the horizontal line test, it must have at least one point on the graph where the slope is zero. one-to-one not one-to-one
Now that we know that we can use the derivative to find the slope of a function, this observation leads to the following theorem: Derivatives of Inverse functions: If f is differentiable at every point of an interval I and df/dx is never zero on I, then f has an inverse and f -1 is differentiable at every point of the interval f(I).
Example: Does have an inverse? Since is never zero, must pass the horizontal line test, so it must have an inverse.
We can use implicit differentiation to find:
We can use implicit differentiation to find: But so is positive.
We could use the same technique to find d sec 1 x. dx and
Your calculator contains all six inverse trig functions. However it is occasionally still useful to know the following: p
Homework: 3. 8 a 3. 8 p 170 3, 12 3. 7 p 162 33, 42, 51 2. 3 p 85 51 3. 8 b 3. 8 p 170 6, 15, 21, 29 2. 4 p 92 5, 23
- Derivatives of arc functions
- Derivative of inverse trig functions
- Derivatives of inverse functions and logarithms
- Range of inverse sine function
- Properties of inverse functions
- Lesson 4 the sine function
- Inverse trig ratios and finding missing angles
- Inverse trig table
- Differentiation of tan inverse x
- Implicit differentiation with inverse trig functions
- Sin inverse formula
- Inverse function derivative
- Evaluating inverse trig functions without a calculator
- Arccos 2