Finding The Second Derivative Of An Integral Function A Calculus Exploration

Hey guys! Today, we're diving into a super interesting problem from the realm of calculus that involves finding the second derivative of a function defined as an integral. It might sound intimidating at first, but trust me, we'll break it down step by step, making it crystal clear. We'll explore the fundamental concepts behind this problem and equip you with the knowledge to tackle similar challenges with confidence. Let's get started!

Okay, so here's the problem we're going to tackle: If ${ f(x) = \int_0^x (t^3 + 3t^2 + 7) dt }$, then what is ${ f''(x) }$?

This question is a classic example of combining the concepts of integration and differentiation. We need to find the second derivative of a function that is itself defined as an integral. This involves a cool interplay between the Fundamental Theorem of Calculus and the rules of differentiation. We're essentially peeling back the layers of the function, first by understanding the integral and then by differentiating it twice.

Before we jump into solving the problem directly, let's quickly recap the key concepts we'll be using. This will ensure we're all on the same page and that the solution makes perfect sense.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is the cornerstone of this problem. It has two parts, but the one we're most interested in right now is the first part. This part essentially tells us how to differentiate an integral. It states that if we have a function defined as:

${ F(x) = \int_a^x f(t) dt }$

where ${ a }$ is a constant, then the derivative of ${ F(x) }$ is simply the integrand evaluated at ${ x }$:

${ F'(x) = f(x) }$

In simpler terms, differentiating an integral "undoes" the integration, giving us back the original function inside the integral, but with the variable of integration, ${ t }$, replaced by ${ x }$. This is a powerful concept and the key to solving our problem.

Differentiation Rules

Of course, we'll also need to remember our basic differentiation rules. Specifically, we'll be using the power rule, which states that if ${ f(x) = x^n }$, then ${ f'(x) = nx^{n-1} }$. This rule will be essential when we differentiate the polynomial we get after applying the Fundamental Theorem of Calculus.

The Chain Rule (Implicitly)

While not explicitly used, it's good to keep in mind that the Chain Rule is implicitly at play here. When we differentiate ${ f(x) = \int_0^x (t^3 + 3t^2 + 7) dt }$, we're essentially differentiating a composite function. The outer function is the integral, and the inner function is the upper limit of integration, which is ${ x }$. However, since the derivative of ${ x }$ with respect to ${ x }$ is just 1, we don't see the Chain Rule explicitly in the calculations.

Okay, now that we've reviewed the key concepts, let's start solving the problem. Our first step is to find the first derivative, ${ f'(x) }$. We'll use the Fundamental Theorem of Calculus for this.

We have:

${ f(x) = \int_0^x (t^3 + 3t^2 + 7) dt }$

Applying the Fundamental Theorem of Calculus, we simply replace ${ t }$ with ${ x }$ in the integrand:

${ f'(x) = x^3 + 3x^2 + 7 }$

See? It's that straightforward! The integral sign disappears, and we're left with a simple polynomial. This is the power of the Fundamental Theorem of Calculus in action. We've successfully found the first derivative of our function.

Now comes the fun part – finding the second derivative, ${ f''(x) }$. This is simply the derivative of ${ f'(x) }$. We already found ${ f'(x) }$ in the previous step, so we just need to differentiate it again.

We have:

${ f'(x) = x^3 + 3x^2 + 7 }$

To find ${ f''(x) }$, we'll differentiate each term with respect to ${ x }$. We'll use the power rule for this.

• The derivative of ${ x^3 }$ is ${ 3x^2 }$ (using the power rule with ${ n = 3 }$).
• The derivative of ${ 3x^2 }$ is ${ 6x }$ (using the power rule with ${ n = 2 }$ and multiplying by the constant 3).
• The derivative of ${ 7 }$ is ${ 0 }$ (the derivative of a constant is always zero).

Putting it all together, we get:

${ f''(x) = 3x^2 + 6x + 0 }$

Simplifying, we have:

${ f''(x) = 3x^2 + 6x }$

And there you have it! We've successfully found the second derivative of the function. It's a quadratic function, which is a common result when differentiating polynomials.

So, the final answer to our problem is:

${ f''(x) = 3x^2 + 6x }$

This is the second derivative of the function ${ f(x) = \int_0^x (t^3 + 3t^2 + 7) dt }$. We arrived at this answer by applying the Fundamental Theorem of Calculus to find the first derivative and then using the power rule to differentiate again.

Let's recap the key takeaways from this problem. This will help solidify your understanding and allow you to apply these concepts to other problems.

• The Fundamental Theorem of Calculus is your friend. It provides a direct way to differentiate functions defined as integrals.
• Differentiation rules are essential. Knowing the power rule and other basic differentiation rules is crucial for finding derivatives.
• Don't be afraid to break down the problem. Complex problems can be solved by breaking them down into smaller, more manageable steps.
• Practice makes perfect. The more you practice these types of problems, the more comfortable you'll become with the concepts.

It's also helpful to be aware of common mistakes that students make when solving these types of problems. Here are a few pitfalls to watch out for:

• Forgetting the Fundamental Theorem of Calculus. This is the most crucial tool for this type of problem. Make sure you understand it thoroughly.
• Making mistakes in differentiation. Double-check your differentiation steps, especially when using the power rule.
• Ignoring the limits of integration. The Fundamental Theorem of Calculus applies when the upper limit of integration is ${ x }$. If it's a more complex function of ${ x }$, you'll need to use the Chain Rule more explicitly.
• Not simplifying the final answer. Always simplify your answer as much as possible.

To really master these concepts, it's essential to practice! Here are a few similar problems you can try:

1. If ${ g(x) = \int_1^x (2t^4 - t^2 + 5) dt }$, find ${ g''(x) }$.
2. If ${ h(x) = \int_0^x (u^3 + 4u) du }$, find ${ h''(x) }$.
3. If ${ k(x) = \int_2^x (z^2 - 6z + 9) dz }$, find ${ k''(x) }$.

Try solving these problems on your own, and you'll be well on your way to mastering the concept of finding the second derivative of an integral function.

So, there you have it! We've successfully navigated the world of finding the second derivative of a function defined as an integral. We've seen how the Fundamental Theorem of Calculus and basic differentiation rules come together to solve this type of problem. Remember, the key is to understand the underlying concepts and practice regularly. Keep exploring, keep learning, and keep pushing your mathematical boundaries!

I hope this explanation was helpful, guys. If you have any questions or want to explore more calculus topics, feel free to ask. Happy learning!