First 15 Dec 4 Review

Front

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

Back

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

Front

Find the derivative of \( y=\frac{x}{\sin 3x} \)

Back

The derivative is \( \frac{\sin 3x - 3x \cos 3x}{(\sin 3x)^2} \).

Front

If g is the inverse function of f and f(2)=3, find g'(3) for f(x)=6x²+2x-1.

Back

g'(3) = \( \frac{1}{f'(2)} \) where \( f'(x) = 12x + 2 \). Thus, \( g'(3) = \frac{1}{26} \).

Tags

CCSS.HSF-BF.B.4A

Front

What is the chain rule in differentiation?

Back

The chain rule states that if a function \( y = f(g(x)) \), then the derivative is \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

Front

Define the product rule in differentiation.

Back

The product rule states that if \( u(x) \) and \( v(x) \) are functions, then \( (uv)' = u'v + uv' \).

Front

What is the quotient rule in differentiation?

Back

The quotient rule states that if \( u(x) \) and \( v(x) \) are functions, then \( \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \).

Front

What is the derivative of e^x?

Back

The derivative of e^x is e^x.

Front

What is the derivative of ln(x)?

Back

The derivative of ln(x) is \( \frac{1}{x} \) for \( x > 0 \).

Front

Explain the concept of inverse functions.

Back

Inverse functions reverse the effect of the original function. If \( f(a) = b \), then \( g(b) = a \) where \( g \) is the inverse of \( f \).

Tags

CCSS.HSF-BF.B.4A

Front

What is the relationship between a function and its inverse regarding derivatives?

Back

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

Tags

CCSS.HSF-BF.B.4A