Derivative & Integration Rules

Front

What is the limit of \( h(x) = \frac{f(x)}{g(x)} \) as \( x \rightarrow \pm\infty \) if the degree of \( g(x) \) is greater than the degree of \( f(x) \)?

Back

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Front

How can you determine if a function \( f \) is increasing on the interval \( (a,b) \)?

Back

If \( f'(x) > 0 \) for all \( x \) in \( (a,b) \).

Front

What is the product rule for differentiation?

Back

\( \frac{d}{dx}[uv] = u'v + uv' \) where \( u \) and \( v \) are functions of \( x \).

Front

What is the quotient rule for differentiation?

Back

\( \frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - vu'}{v^2} \) where \( u \) and \( v \) are functions of \( x \).

Front

What is the chain rule for differentiation?

Back

\( \frac{d}{dx}[f(g(x))] = f'(g(x))g'(x) \) where \( f \) and \( g \) are functions of \( x \).

Front

What does it mean if \( f''(x) > 0 \)?

Back

The function \( f \) is concave up on that interval.

Front

What does it mean if \( f''(x) < 0 \)?

Back

The function \( f \) is concave down on that interval.

Front

What is the Fundamental Theorem of Calculus?

Back

If \( F \) is an antiderivative of \( f \) on \( [a,b] \), then \( \int_a^b f(x)dx = F(b) - F(a) \).

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 \( \frac{1}{x} \).