6.6.1 Derivatives of Inverse Trigonometric Functions Flashcards
1
Q
Derivatives of Inverse Trigonometric Functions
A
• To find the derivative of an inverse trig function, rewrite the expression in terms of standard trig functions, differentiate implicitly, and use the Pythagorean theorem.
2
Q
note
A
- To find the derivative of arcsin x, first think of it as
y = arcsin x. Then rewrite the expression using standard
trigonometric functions. - Use implicit differentiation to take the derivative of both
sides. - Remember to use the Chain Rule. The derivative of y with respect to x is dy/dx.
- The result you get by differentiating is in terms of y, but you want it in terms of x.
- Let y be an angle of a right triangle. Since sin y = x, you can let the opposite side equal x and the hypotenuse equal one.
- Using the Pythagorean theorem, you can write cos y in terms of x. Now you have an expression for the derivative of arcsin x.
- It may seem strange that the derivative is not in terms of any of the other trig or inverse trig functions. If you remember that the trigonometric functions are all defined by right triangles, then the derivative makes more sense.
- If you use the same method as above, you can determine the derivatives of all of the inverse trig functions.
- These derivatives may be difficult to memorize. But if you remember the method, then you can always derive them again.
3
Q
Find dy/dx for y=arcsecx
A
1/|x|√x^2−1
4
Q
On which of the following intervals is tan−1 x increasing?
A
(−∞, ∞)
5
Q
On which of the following intervals is arctan x concave down?
A
(0, ∞)
6
Q
Find dy/dx for y=arccscx
A
−1/|x|√x^2−1
7
Q
Find dy/dx for y=arccosx.
A
−1/√1−x^2
8
Q
On which of the following intervals is sin−1 x increasing?
A
(−1, 1)
9
Q
Find dy/dx for y=arcsin x
A
1/√1−x^2
10
Q
Which of the following represents the interval on which cos−1 x is concave up?
A
(−1, 0)
11
Q
Find dy/dx for y = arctanx.
A
1/1+x^2
12
Q
Find dy/dx for y=arccotx
A
-1/1+x^2
