Calculus Derivatives and Notation

Front

What is a derivative?

Back

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental concept in calculus that measures how a function's output changes as its input changes.

Front

What is the notation for the derivative of a function f(x)?

Back

The derivative of a function f(x) is denoted as f'(x) or \frac{df}{dx}.

Front

What does it mean for a function to be increasing?

Back

A function is increasing on an interval if its derivative is positive throughout that interval.

Front

What does it mean for a function to be decreasing?

Back

A function is decreasing on an interval if its derivative is negative throughout that interval.

Front

What is the Power Rule for derivatives?

Back

The Power Rule states that if f(x) = x^n, then f'(x) = n*x^{n-1}.

Front

What is the product rule for derivatives?

Back

The product rule states that if u(x) and v(x) are functions, then the derivative of their product is given by (uv)' = u'v + uv'.

Front

What is the quotient rule for derivatives?

Back

The quotient rule states that if u(x) and v(x) are functions, then the derivative of their quotient is given by (u/v)' = (u'v - uv')/v^2.

Front

What is the significance of a horizontal tangent line?

Back

A horizontal tangent line indicates that the derivative of the function at that point is zero, meaning the function has a local maximum or minimum.

Front

What is the Chain Rule for derivatives?

Back

The Chain Rule states that if a function y = f(g(x)), then the derivative is given by dy/dx = f'(g(x)) * g'(x).

Front

What is the definition of a critical point?

Back

A critical point occurs where the derivative of a function is either zero or undefined, indicating potential local maxima or minima.