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AP Calculus AB: 6.6.1 Derivatives of Inverse Trigonometric Functions

Mathematics12 CardsCreated 3 months ago

This section explains how to derive and understand the formulas for the derivatives of inverse trigonometric functions using implicit differentiation and right triangle definitions. It emphasizes applying the Chain Rule and the Pythagorean Theorem to rewrite results in terms of π‘₯ making it easier to differentiate expressions involving arcsin, arccos, arctan, and other inverse trig functions.

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Derivatives of Inverse Trigonometric Functions

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.

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Key Terms

Term
Definition

Derivatives of Inverse Trigonometric Functions

To find the derivative of an inverse trig function, rewrite the expression in terms of standard trig functions, differentiate implicitly, and use t...

note

  • To find the derivative of arcsin x, first think of it as
    y = arcsin x. Then rewrite the expression using standard
    trigonometric funct...

Find dy/dx for y=arcsecx

1/|x|√x^2βˆ’1

On which of the following intervals is tanβˆ’1 x increasing?

(βˆ’βˆž, ∞)

On which of the following intervals is arctan x concave down?

(0, ∞)

Find dy/dx for y=arccscx

βˆ’1/|x|√x^2βˆ’1

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AP Calculus AB: 6.6.1 Derivatives of Inverse Trigonometric Functions
TermDefinition

Derivatives of Inverse Trigonometric Functions

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.

note

  • 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.

Find dy/dx for y=arcsecx

1/|x|√x^2βˆ’1

On which of the following intervals is tanβˆ’1 x increasing?

(βˆ’βˆž, ∞)

On which of the following intervals is arctan x concave down?

(0, ∞)

Find dy/dx for y=arccscx

βˆ’1/|x|√x^2βˆ’1

Find dy/dx for y=arccosx.

βˆ’1/√1βˆ’x^2

On which of the following intervals is sinβˆ’1 x increasing?

(βˆ’1, 1)

Find dy/dx for y=arcsin x

1/√1βˆ’x^2

Which of the following represents the interval on which cosβˆ’1 x is concave up?

(βˆ’1, 0)

Find dy/dx for y = arctanx.

1/1+x^2

Find dy/dx for y=arccotx

-1/1+x^2