Derivatives

Front

What is a derivative?

Back

A derivative represents the rate of change of a function with respect to a variable. It is a measure of how a function's output value changes as its input value changes.

Front

What is the derivative of sin(x)?

Back

The derivative of sin(x) is cos(x).

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

How do you find the derivative of a constant?

Back

The derivative of a constant is 0.

Front

What is the product rule for derivatives?

Back

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

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 chain rule for derivatives?

Back

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

Front

Find the derivative of e^(3x).

Back

The derivative of e^(3x) is 3e^(3x).

Front

What does it mean for a function to be continuous but not differentiable?

Back

A function is continuous but not differentiable at a point if it has a sharp corner or cusp at that point, meaning it does not have a defined tangent.

Front

What is the instantaneous velocity of a particle?

Back

The instantaneous velocity is the derivative of the position function with respect to time, evaluated at a specific time.