Basic Derivative Rules

Front

What is the Quotient Rule for derivatives?

Back

If \( f \) and \( g \) are functions, then the derivative of their quotient is given by: \( \frac{d}{dx}\left[\frac{f}{g}\right] = \frac{g \cdot f' - f \cdot g'}{g^2} \).

Front

What is the derivative of \( \arcsin(u) \)?

Back

The derivative of \( \arcsin(u) \) is given by: \( \frac{d}{dx}[\arcsin(u)] = \frac{1}{\sqrt{1-u^2}} \cdot du \).

Front

What is the Power Rule for derivatives?

Back

The Power Rule states that if \( n \) is a constant, then \( \frac{d}{dx}[x^n] = n \cdot x^{n-1} \).

Front

What is the derivative of \( \sqrt{x} \)?

Back

The derivative of \( \sqrt{x} \) is given by: \( \frac{d}{dx}[\sqrt{x}] = \frac{1}{2}x^{-\frac{1}{2}} \).

Front

What is the derivative of \( \arccos(u) \)?

Back

The derivative of \( \arccos(u) \) is given by: \( \frac{d}{dx}[\arccos(u)] = \frac{-1}{\sqrt{1-u^2}} \cdot du \).

Front

What is the Chain Rule for derivatives?

Back

The Chain Rule states that if \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

Front

What is the derivative of a constant function?

Back

The derivative of a constant function is zero: \( \frac{d}{dx}[c] = 0 \) where \( c \) is a constant.

Front

What is the derivative of \( e^x \)?

Back

The derivative of \( e^x \) is \( \frac{d}{dx}[e^x] = e^x \).

Front

What is the derivative of \( \ln(x) \)?

Back

The derivative of \( \ln(x) \) is \( \frac{d}{dx}[\ln(x)] = \frac{1}{x} \).

Front

What is the product rule for derivatives?

Back

If \( u \) and \( v \) are functions, then the product rule states: \( \frac{d}{dx}[uv] = u'v + uv' \).