Inverse Trig Derivatives

Front

What is the derivative of \( \sin^{-1}(x) \)?

Back

The derivative is \( \frac{1}{\sqrt{1-x^2}} \) for \( -1 < x < 1 \).

Front

What is the derivative of \( \cos^{-1}(x) \)?

Back

The derivative is \( -\frac{1}{\sqrt{1-x^2}} \) for \( -1 < x < 1 \).

Front

What is the derivative of \( \tan^{-1}(x) \)?

Back

The derivative is \( \frac{1}{1+x^2} \) for all \( x \).

Front

What is the derivative of \( \cot^{-1}(x) \)?

Back

The derivative is \( -\frac{1}{1+x^2} \) for all \( x \).

Front

What is the derivative of \( \sec^{-1}(x) \)?

Back

The derivative is \( \frac{1}{|x|\sqrt{x^2-1}} \) for \( |x| > 1 \).

Front

What is the derivative of \( \csc^{-1}(x) \)?

Back

The derivative is \( -\frac{1}{|x|\sqrt{x^2-1}} \) for \( |x| > 1 \).

Front

What is the relationship between inverse trigonometric functions and their derivatives?

Back

The derivatives of inverse trigonometric functions involve square roots and rational functions, often requiring the use of the Pythagorean identity.

Front

How do you find the derivative of \( \sin^{-1}(x) \) using implicit differentiation?

Back

Let \( y = \sin^{-1}(x) \), then \( x = \sin(y) \). Differentiate both sides to find \( \frac{dx}{dy} = \cos(y) \) and use \( \cos^2(y) + \\sin^2(y) = 1 \) to express \( \frac{dy}{dx} \).

Front

What is the domain of \( \sin^{-1}(x) \)?

Back

The domain is \( [-1, 1] \).

Front

What is the range of \( \tan^{-1}(x) \)?

Back

The range is \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).