Topics 3.3 and 3.4 Derivatives of Inverses and Inverse Trig

Front

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

Back

The derivative is \( \frac{d}{dx} \left[ \sin^{-1}(x) \right] = \frac{1}{\sqrt{1 - x^2}} \) for \( -1 < x < 1 \).

Front

What is the formula for the derivative of the inverse cosine function, \( \arccos(x) \)?

Back

The derivative is \( \frac{d}{dx} \left[ \arccos(x) \right] = -\frac{1}{\sqrt{1 - x^2}} \) for \( -1 < x < 1 \).

Front

How do you find the derivative of \( \sin^{-1}(u) \) where \( u \) is a function of \( x \)?

Back

Use the chain rule: \( \frac{d}{dx} \left[ \sin^{-1}(u) \right] = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \).

Front

What is the derivative of \( \arccos(u) \) where \( u \) is a function of \( x \)?

Back

Use the chain rule: \( \frac{d}{dx} \left[ \arccos(u) \right] = -\frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \).

Front

What is the relationship between the derivative of a function and its inverse?

Back

If \( y = f^{-1}(x) \), then \( \frac{dy}{dx} = \frac{1}{f'(f^{-1}(x))} \).

Front

How do you find the derivative of an inverse function at a specific point?

Back

Use the formula: \( g'(f(a)) = \frac{1}{f'(a)} \) where \( g \) is the inverse of \( f \).

Front

What is the derivative of \( f(x) = 6x^2 + 2x - 1 \)?

Back

The derivative is \( f'(x) = 12x + 2 \).

Front

If \( f(2) = 3 \) for the function \( f(x) \), how do you find \( g'(3) \) where \( g \) is the inverse of \( f \)?

Back

Use the formula: \( g'(3) = \frac{1}{f'(2)} \). Calculate \( f'(2) \) first.

Front

What is the derivative of \( g(x) = x^5 + 3x \)?

Back

The derivative is \( g'(x) = 5x^4 + 3 \).

Front

If \( g(1) = 4 \), how do you find \( h'(4) \) where \( h \) is the inverse of \( g \)?

Back

Use the formula: \( h'(4) = \frac{1}{g'(1)} \). Calculate \( g'(1) \) first.