Higher Order Derivatives

Summary

This video tutorial focuses on finding higher-order derivatives of functions. It begins with a polynomial function, demonstrating the process of calculating first and second derivatives using the power rule. The tutorial then explores the product rule through a function involving both polynomial and trigonometric components, detailing the steps to obtain the second derivative. Lastly, it addresses finding the third and fourth derivatives from given functions, highlighting the importance of rewriting expressions for clarity. Overall, the video provides essential techniques for effectively computing derivatives in calculus.

Takeaways

• 😀 The video focuses on finding higher-order derivatives of functions.
• 😀 To find the second derivative, start by calculating the first derivative using the power rule.
• 😀 The power rule states that the derivative of x^n is n*x^(n-1).
• 😀 Example 1: For the function f(x) = 3x^5 + 2x^3 - 6x + 4, the first derivative is f'(x) = 15x^4 + 6x^2 - 6.
• 😀 The second derivative of the same function is f''(x) = 60x^3 + 12x.
• 😀 When dealing with products of functions, use the product rule to find derivatives.
• 😀 Example 2: For h(x) = x^2 * cos(x), the first derivative involves applying the product rule.
• 😀 The second derivative of h(x) combines derivatives of both terms and simplifies to h''(x) = 2cos(x) - 4xsin(x) - x^2cos(x).
• 😀 Higher-order derivatives can be computed from existing derivatives, such as finding the third and fourth derivatives from the second derivative.
• 😀 Example 3 illustrates finding the third derivative for f(x) = sqrt(x) by rewriting it as x^(1/2) and applying the power rule repeatedly.

Q & A

What is the first step to find the second derivative of the function f(x) = 3x^5 + 2x^3 - 6x + 4?

The first step is to find the first derivative by differentiating the function using the power rule.

How do you apply the power rule to differentiate x^5?

Using the power rule, the derivative of x^5 is calculated as 5x^(5-1), which simplifies to 5x^4.

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

The first derivative is f'(x) = 15x^4 + 6x^2 - 6.

What is the formula for the product rule when differentiating a product of two functions?

The product rule states that if you have two functions f and g, then the derivative of their product is f' g + f g'.

How do you find the first derivative of the function h(x) = x^2 cos(x)?

Using the product rule, the first derivative is h'(x) = 2x cos(x) - x^2 sin(x).

What is the second derivative of h(x) = x^2 cos(x)?

The second derivative is h''(x) = 2 cos(x) - 4x sin(x) - x^2 cos(x).

How do you express the square root of x in terms of a power for differentiation?

The square root of x can be expressed as x^(1/2) for differentiation purposes.

What is the third derivative of f(x) = sqrt(x)?

The third derivative is f'''(x) = (3/8)x^(-5/2) or 3/(8sqrt(x^5)).

What should you do if given a second derivative like 5/x^2 to find the fourth derivative?

You need to differentiate the second derivative two more times to reach the fourth derivative.

What is the final form of the fourth derivative if the second derivative is f''(x) = 5/x^2?

The fourth derivative is f^{(4)}(x) = 30/x^4 or 30x^{-4}.