3.4 Derivatives of Inverses and Inverse Trig

Front

What is the derivative of the inverse function at a point?

Back

If \( h \) is the inverse of \( g \), then \( h'(g(a)) = \frac{1}{g'(a)} \).

Tags

CCSS.HSF-BF.B.4A

Front

Find the derivative of \( y = \arccos(x) \).

Back

\( \frac{dy}{dx} = -\frac{1}{\sqrt{1-x^2}} \) for \( -1 < x < 1 \).

Tags

CCSS.HSF.TF.B.7

Front

What is the relationship between the derivatives of inverse functions?

Back

The derivative of an inverse function is the reciprocal of the derivative of the original function evaluated at the corresponding point.

Tags

CCSS.HSF-BF.B.4A

Front

Find \( f'(4) \) if \( f(x) = -4x + 16 \).

Back

\( f'(4) = -4 \).

Tags

CCSS.HSF.IF.A.2

Front

Given \( y = \arccos(5x^5) \), find \( \frac{dy}{dx} \).

Back

\( \frac{dy}{dx} = \frac{-25x^4}{\sqrt{1-25x^{10}}} \).

Front

If \( g(x) = x^5 + 3x \), find \( g'(x) \).

Back

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

Front

What is the derivative of \( y = \arcsin(x) \)?

Back

\( \frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}} \) for \( -1 < x < 1 \).

Front

Find \( h'(4) \) if \( g(1) = 4 \) and \( g'(1) = 8 \).

Back

\( h'(4) = \frac{1}{g'(1)} = \frac{1}{8} \).

Tags

CCSS.HSF-BF.A.1B

Front

What is the derivative of \( y = \arctan(x) \)?

Back

\( \frac{dy}{dx} = \frac{1}{1+x^2} \).

Front

If \( g(x) = 2x^3 + 3x \), find \( g'(x) \).

Back

\( g'(x) = 6x^2 + 3 \).

Tags

CCSS.HSF.IF.A.2