Derivatives

Front

What is the definition of a derivative?

Back

A derivative represents the rate of change of a function with respect to a variable. It is the limit of the average rate of change of the function as the interval approaches zero.

Front

What is the instantaneous velocity function?

Back

The instantaneous velocity function is the derivative of the position function with respect to time, representing the velocity of an object at a specific moment.

Front

How do you find the derivative of a polynomial function?

Back

To find the derivative of a polynomial function, apply the power rule: for each term ax^n, the derivative is n*ax^(n-1).

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 relationship between velocity and acceleration?

Back

Acceleration is the derivative of velocity with respect to time, indicating how velocity changes over time.

Front

Find the derivative of f(x) = x^4 + 4x^3 - 2x^2.

Back

f'(x) = 4x^3 + 12x^2 - 4x.

Front

Find the derivative of f(x) = (-2/3)x^3 - (1/2)x^2 + 9x.

Back

f'(x) = -2x^2 - x + 9.

Front

What is the definition of acceleration?

Back

Acceleration is the rate of change of velocity with respect to time, often represented as the second derivative of the position function.

Front

How do you calculate the acceleration from a velocity function?

Back

To calculate acceleration, take the derivative of the velocity function with respect to time.

Front

What is the derivative of a constant?

Back

The derivative of a constant is zero.