Higher Order Derivatives

Front

What is a higher order derivative?

Back

A higher order derivative is the derivative of a derivative. For example, the second derivative is the derivative of the first derivative.

Front

How do you find the second derivative of a function?

Back

To find the second derivative, first find the first derivative of the function, then differentiate that result.

Front

What is the second derivative of the function y = 3x^2 + 5x - 1?

Back

The second derivative is 6.

Front

What is the second derivative of f(x) = cos(5x^4)?

Back

The second derivative is f''(x) = -120x^5 cos(5x^4) - 20x^3 sin(5x^4).

Front

What is the second derivative of f(x) = x^2 + e^x - cos(x)?

Back

The second derivative is f''(x) = 2 + e^x + cos(x).

Front

What is the second derivative of y = ln(x^3√x)?

Back

The second derivative is -7/2x^2.

Front

How do you denote the third derivative of a function?

Back

The third derivative is denoted as f'''(x) or d^3y/dx^3.

Front

What is the third derivative of f(x) = 10x^5 + 3x^4 - 5x^3 + 2x^2 - 10x + 6?

Back

The third derivative is f'''(x) = 600x^2 + 72x - 30.

Front

What is the relationship between the first, second, and third derivatives?

Back

The first derivative represents the rate of change, the second derivative represents the rate of change of the rate of change (acceleration), and the third derivative represents the rate of change of the acceleration.

Front

What does it mean if the second derivative is positive?

Back

If the second derivative is positive, the function is concave up, indicating that the slope of the function is increasing.