Problem 38 - Calculus for Biology and Medicine

Question

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{1+x^{2}}^{x^{3}-2 x}(t+1) d t $$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = 3x^5 - 8x^3 + 3x^2 - 2x - 2.\)

Step 1: Understand the Problem

We need to differentiate the given function with respect to \(x\). The function is an integral with variable limits of integration, \(y = \int_{1+x^2}^{x^3 - 2x} (t + 1) \, dt\).

Step 2: Use Leibniz's Rule

Leibniz's rule allows us to differentiate under the integral sign when the limits of integration are functions of \(x\). The derivative \(\frac{dy}{dx}\) can be calculated by differentiating the integral as follows:\[\frac{dy}{dx} = (F(x^3 - 2x) \cdot (3x^2 - 2)) - (F(1+x^2) \cdot (2x))\]where \(F(t) = \int(t + 1) \, dt\), thus, \(F'(t) = t + 1\).

Step 3: Calculate the Antiderivative

Find the antiderivative of the integrand \((t + 1)\) before applying Leibniz's Rule. Since the antiderivative of \(t+1\) is \(F(t) = \frac{t^2}{2} + t + C\), by the Fundamental Theorem of Calculus:\[F(t)' = t + 1.\]

Step 4: Apply Leibniz's Rule

Substitute the results from the previous steps into Leibniz's rule:\[\frac{dy}{dx} = ((x^3 - 2x + 1)\cdot (3x^2 - 2)) - ((1 + x^2 + 1) \cdot (2x)).\]

Step 5: Simplify the Expression

Simplify the expression obtained:\[\frac{dy}{dx} = ((x^3 - 2x + 1)(3x^2 - 2)) - ((2 + x^2) \cdot 2x)\]Distribute and combine terms:\[\frac{dy}{dx} = (3x^5 - 2x^3 + 3x^2 - 6x^3 + 4x - 2) - (4x + 2x^3)\]Combine all like terms:\[\frac{dy}{dx} = 3x^5 - 8x^3 + 3x^2 + 4x - 6x - 2.\]Thus, simplifying further gives:\[\frac{dy}{dx} = 3x^5 - 8x^3 + 3x^2 - 2x - 2.\]

Step 6: Conclusion: Final Answer

After simplification, the derivative \(\frac{dy}{dx}\) of the given function is:\[\frac{dy}{dx} = 3x^5 - 8x^3 + 3x^2 - 2x - 2.\]

Key Concepts

Leibniz's rule

In calculus, Leibniz's rule is a powerful tool for differentiating integrals when the limits of integration are functions of a variable.
Here, we encounter this precisely when differentiating an integral like \( y = \int_{1+x^2}^{x^3-2x}(t+1) \, dt \).

Let's break it down:

• Leibniz's rule allows differentiation under the integral sign.
• This rule is especially helpful for integrals where the limits themselves are functions of the variable we are differentiating with respect to.

The formula is given by:
\[\frac{dy}{dx} = \frac{d}{dx} \left( \int_{a(x)}^{b(x)} f(t) \, dt \right) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)\]
In our example,

• \(a(x) = 1+x^2\)
• \(b(x) = x^3-2x\)
• \(f(t) = t+1\)

By applying Leibniz's rule, we calculate the derivative of the integral by considering both the upper and lower boundaries separately with respect to \(x\).
Finally, we put it together to find \(\frac{dy}{dx}\).

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation with integration.
It provides a concise way to evaluate the integral of a function and also explains how integration and differentiation are inverse processes.

Specifically, the theorem has two parts:

• The first part helps evaluate definite integrals using antiderivatives.
• The second part enables us to differentiate an integral.

For the problem at hand:
To find \( \frac{dy}{dx} \), we recognize \( \int(t + 1) \, dt \) and solve for its antiderivative:
\[F(t) = \frac{t^2}{2} + t + C\]
Concretely, if \( F'(t) = t + 1 \), then the Fundamental Theorem implies:
\[\int (t + 1) \, dt = \frac{t^2}{2} + t + C\]

This helps us differentiate under the integral sign, as we'll know exactly what function \( F(t) \) we need to differentiate.
The theorem reminds us that evaluating \( F(t) \) at the new boundaries is crucial in Leibniz’s rule.

Differentiation under the integral sign

Differentiation under the integral sign is another key aspect of Leibniz’s rule.
It's a technique that involves taking the derivative of an integral without explicitly evaluating it first.
This method might sound complex, but it's quite straightforward once you understand the principles.

When encountering an integral with variable limits, use these strategies:

• Focus on the variable limits of integration.
• Apply Leibniz’s rule, keeping both limits of the integral in mind.
• Combine the results by subtracting the derivative evaluated at the lower limit from the upper limit.

For example, consider \( \int_{1+x^2}^{x^3-2x}(t+1) \, dt \).
We apply differentiation under the integral sign by considering what changes as \( x \) varies.
This requires careful attention: differentiate both upper and lower limits while also accounting for the derivative of the function within the integrand.
Ultimately, this gives us the complete derivative \( \frac{dy}{dx} \), effectively showcasing the seamless workflow of Leibniz's rule.