Simplifying the Second Derivative

Oct 13, 2025 by windailys.com 102 views

Introduction

In calculus, finding the derivative of a function is a crucial concept that helps us understand the behavior of the function. However, when dealing with complex functions, finding the derivative can be a challenging task. In this article, we will explore how to find the second derivative of a given function, which is a crucial concept in calculus.

The Given Function

The given function is:

${ H(x) = \left(x^{12} + 3x^{11}\right) \left(\sqrt{x^{12} + 3x^{11}}\right) + 13x^5 }$

This function is a combination of two terms, where the first term is a product of two expressions, and the second term is a simple power function.

Step 1: Find the First Derivative

To find the second derivative, we need to start by finding the first derivative of the function. We can use the product rule and the chain rule to find the derivative.

Using the product rule, we can write:

${ H^{\prime}(x) = \frac{d}{dx} \left(x^{12} + 3x^{11}\right) \left(\sqrt{x^{12} + 3x^{11}}\right) + \frac{d}{dx} (13x^5) }$

Applying the chain rule, we get:

${ H^{\prime}(x) = \left(x^{12} + 3x^{11}\right) \frac{d}{dx} \left(\sqrt{x^{12} + 3x^{11}}\right) + \left(\sqrt{x^{12} + 3x^{11}}\right) \frac{d}{dx} \left(x^{12} + 3x^{11}\right) + 65x^4 }$

Simplifying the expression, we get:

${ H^{\prime}(x) = \left(x^{12} + 3x^{11}\right) \frac{1}{2} \frac{1}{\sqrt{x^{12} + 3x^{11}}} \left(12x^{11} + 33x^{10}\right) + \left(\sqrt{x^{12} + 3x^{11}}\right) \left(12x^{11} + 33x^{10}\right) + 65x^4 }$

Combining like terms, we get:

${ H^{\prime}(x) = \frac{1}{2} \left(12x^{11} + 33x^{10}\right) \left(x^{12} + 3x^{11}\right) \frac{1}{\sqrt{x^{12} + 3x^{11}}} + \left(\sqrt{x^{12} + 3x^{11}}\right) \left(12x^{11} + 33x^{10}\right) + 65x^4 }$

Simplifying further, we get:

${ H^{\prime}(x) = \frac{1}{2} \left(12x^{11} + 33x^{10}\right) \left(x^{12} + 3x^{11}\right) \frac{1}{\sqrt{x^{12} + 3x^{11}}} + \left(12x^{11} + 33x^{10}\right) \sqrt{x^{12} + 3x^{11}} + 65x^4 }$

Step 2: Find the Second Derivative

To find the second derivative, we need to differentiate the first derivative with respect to x.

Using the product rule and the chain rule, we can write:

${ H^{\prime \prime}(x) = \frac{d}{dx} \left(\frac{1}{2} \left(12x^{11} + 33x^{10}\right) \left(x^{12} + 3x^{11}\right) \frac{1}{\sqrt{x^{12} + 3x^{11}}} + \left(12x^{11} + 33x^{10}\right) \sqrt{x^{12} + 3x^{11}} + 65x^4\right) }$

Applying the product rule and the chain rule, we get:

${ H^{\prime \prime}(x) = \frac{1}{2} \left(12x^{11} + 33x^{10}\right) \frac{d}{dx} \left(x^{12} + 3x^{11}\right) \frac{1}{\sqrt{x^{12} + 3x^{11}}} + \left(x^{12} + 3x^{11}\right) \frac{d}{dx} \left(\frac{1}{2} \left(12x^{11} + 33x^{10}\right) \frac{1}{\sqrt{x^{12} + 3x^{11}}}\right) + \left(12x^{11} + 33x^{10}\right) \frac{d}{dx} \left(\sqrt{x^{12} + 3x^{11}}\right) + 260x^3 }$

Simplifying the expression, we get:

${ H^{\prime \prime}(x) = \frac{1}{2} \left(12x^{11} + 33x^{10}\right) \left(12x^{11} + 33x^{10}\right) \frac{1}{\sqrt{x^{12} + 3x^{11}}} + \left(x^{12} + 3x^{11}\right) \left(\frac{1}{2} \left(12x^{11} + 33x^{10}\right) \frac{1}{\sqrt{x^{12} + 3x^{11}}}\right) \left(12x^{11} + 33x^{10}\right) + \left(12x^{11} + 33x^{10}\right) \left(\frac{1}{2} \left(12x^{11} + 33x^{10}\right) \frac{1}{\sqrt{x^{12} + 3x^{11}}}\right) + 260x^3 }$

Combining like terms, we get:

${ H^{\prime \prime}(x) = \frac{1}{2} \left(12x^{11} + 33x^{10}\right)^2 \frac{1}{\sqrt{x^{12} + 3x^{11}}} + \left(x^{12} + 3x^{11}\right) \left(\frac{1}{2} \left(12x^{11} + 33x^{10}\right)^2 \frac{1}{\sqrt{x^{12} + 3x^{11}}}\right) + \left(12x^{11} + 33x^{10}\right)^2 \frac{1}{\sqrt{x^{12} + 3x^{11}}} + 260x^3 }$

Simplifying further, we get:

Simplifying the Second Derivative

We can simplify the second derivative by combining like terms.

Combining like terms, we get:

${ H^{\prime \prime}(x) = \frac{3}{2} \left(12x^{11} + 33x^{10}\right)^2 \frac{1}{\sqrt{x^{12} + 3x^{11}}} + 260x^3 }$

Simplifying further, we get:

${ H^{\prime \prime}(x) = \frac{3}{2} \left(132x^{10} + 330x^9\right) \left(x^{12} + 3x^{11}\right) \frac{1}{\sqrt{x^{12} + 3x^{11}}} + 260x^3 }$

Conclusion

Q: What is the second derivative of the given function?

A: The second derivative of the given function is:

${ H^{\prime \prime}(x) = \frac{3}{2} \left(132x^{10} + 330x^9\right) \left(x^{12} + 3x^{11}\right) \frac{1}{\sqrt{x^{12} + 3x^{11}}} + 260x^3 }$

Q: How do I find the second derivative of a complex function?

A: To find the second derivative of a complex function, you need to follow these steps:

Find the first derivative of the function using the product rule and the chain rule.
Differentiate the first derivative with respect to x to find the second derivative.

Q: What is the product rule and the chain rule?

A: The product rule is a rule in calculus that helps us find the derivative of a product of two functions. It states that if we have two functions f(x) and g(x), then the derivative of their product is:

${ \frac{d}{dx} (f(x)g(x)) = f^{\prime}(x)g(x) + f(x)g^{\prime}(x) }$

The chain rule is a rule in calculus that helps us find the derivative of a composite function. It states that if we have a function f(x) and a composite function g(f(x)), then the derivative of the composite function is:

${ \frac{d}{dx} (g(f(x))) = g^{\prime}(f(x)) \cdot f^{\prime}(x) }$

Q: How do I apply the product rule and the chain rule?

A: To apply the product rule and the chain rule, you need to follow these steps:

Identify the two functions that you want to multiply together.
Find the derivatives of each function using the power rule and the sum rule.
Apply the product rule to find the derivative of the product of the two functions.
Apply the chain rule to find the derivative of the composite function.

Q: What is the power rule and the sum rule?

A: The power rule is a rule in calculus that helps us find the derivative of a power function. It states that if we have a function f(x) = x^n, then the derivative of the function is:

${ f^{\prime}(x) = nx^{n-1} }$

The sum rule is a rule in calculus that helps us find the derivative of a sum of two functions. It states that if we have two functions f(x) and g(x), then the derivative of their sum is:

${ \frac{d}{dx} (f(x) + g(x)) = f^{\prime}(x) + g^{\prime}(x) }$

Q: How do I find the derivative of a power function?

A: To find the derivative of a power function, you need to follow these steps:

Identify the power function that you want to find the derivative of.
Apply the power rule to find the derivative of the function.

Q: How do I find the derivative of a sum of two functions?

A: To find the derivative of a sum of two functions, you need to follow these steps:

Identify the two functions that you want to find the derivative of.
Find the derivatives of each function using the power rule and the sum rule.
Apply the sum rule to find the derivative of the sum of the two functions.

Conclusion

In this article, we have answered some common questions about finding the second derivative of a complex function. We have also provided some tips and tricks for applying the product rule and the chain rule. We hope that this article has been helpful in understanding the concept of finding the second derivative of a complex function.