Derivatives Chapter 3

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 power rule for differentiation?

Back

The power rule states that if f(x) = x^n, then f'(x) = n*x^(n-1), where n is a constant.

Front

How do you find the maximum height of a projectile using derivatives?

Back

To find the maximum height, take the derivative of the height function, set it to zero, and solve for time. This gives the time at which the maximum height occurs.

Front

What is the second derivative test?

Back

The second derivative test is used to determine the concavity of a function and to identify local maxima and minima. If f''(x) > 0, the function is concave up (local minimum); if f''(x) < 0, it is concave down (local maximum).

Front

What is the relationship between position, velocity, and acceleration?

Back

Velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity with respect to time.

Front

How do you find the slope of a curve at a given point?

Back

To find the slope of a curve at a specific point, calculate the derivative of the function and evaluate it at that point.

Front

What is the chain rule in differentiation?

Back

The chain rule states that if you have a composite function f(g(x)), then the derivative is f'(g(x)) * g'(x).

Front

What is the product rule for differentiation?

Back

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

Front

What is the quotient rule for differentiation?

Back

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

Front

How do you determine the acceleration from a position function?

Back

To determine acceleration from a position function s(t), take the second derivative of s(t) with respect to time.