Inverse Trig Derivatives

Front

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

Back

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

Front

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

Back

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

Front

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

Back

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

Front

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

Back

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

Front

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

Back

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

Front

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

Back

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

Front

How do you find the derivative of a composite function involving inverse trigonometric functions?

Back

Use the chain rule along with the derivatives of the inverse functions.

Front

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

Back

The derivatives of inverse functions are the reciprocals of the derivatives of their corresponding trigonometric functions.

Front

If \( f(x) = \sin^{-1}(g(x)) \), what is \( f'(x) \)?

Back

Use the chain rule: \( f'(x) = \frac{g'(x)}{\sqrt{1-g(x)^2}} \).

Front

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

Back

The second derivative is \( \frac{x}{(1-x^2)^{3/2}} \) for \( -1 < x < 1 \).