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3.8 Derivatives of Inverse Trig Functions

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Presentation on theme: "3.8 Derivatives of Inverse Trig Functions"— Presentation transcript:

1 3.8 Derivatives of Inverse Trig Functions

2 At x = 2: We can find the inverse function as follows: To find the derivative of the inverse function: Switch x and y.

3 Slopes are reciprocals.
At x = 2: At x = 4:

4 Slopes are reciprocals.
Because x and y are reversed to find the reciprocal function, the following pattern always holds: The derivative of Derivative Formula for Inverses: evaluated at is equal to the reciprocal of the derivative of evaluated at

5 A typical problem using this formula might look like this:
Given: Find: Derivative Formula for Inverses:

6 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

7 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).

8 Example: Does have an inverse?
Since is never zero, must pass the horizontal line test, so it must have an inverse.

9 We can use implicit differentiation to find:

10 We can use implicit differentiation to find:
But so is positive.

11 We could use the same technique to find and
. 1 sec d x dx -

12 p Your calculator contains all six inverse trig functions.
However it is occasionally still useful to know the following: p

13 Homework: 3.8a 3.8 p170 3,12 3.7 p162 33,42,51 2.3 p85 51 3.8b 3.8 p170 6,15,21,29 2.4 p92 5,23


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