Problem 15 - Calculus for Biology and Medicine

Question

Use Leibniz's rule to find \(\frac{d y}{d x}\). $$ y=\int_{0}^{3 x}\left(1+t^{2}\right) d t $$

Step-by-Step Solution

Answer

The derivative is \(3 + 27x^2\).

Step 1: Identify the Leibniz Rule

Leibniz's Rule for differentiation under the integral sign states:\[ \frac{d}{dx}\int_{a(x)}^{b(x)} f(t, x) \, dt = f(b(x), x)\cdot b'(x) - f(a(x), x)\cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(t, x) \, dt \]In this problem, we have constant lower limit and a variable upper limit function.

Step 2: Differentiate Upper Limit

Since the problem features a constant lower limit and a variable upper limit of integration, apply the Leibniz rule directly: \[ \frac{d}{dx} \int_{0}^{3x} (1 + t^2) \, dt = (1 + (3x)^2) \cdot \frac{d}{dx}(3x) \] Differentiate the upper limit function \(3x\), which gives \(3\).

Step 3: Substitute and Simplify

Substitute back into the formula: \[ (1 + (3x)^2) \times 3 = (1 + 9x^2) \times 3 = 3 + 27x^2 \] This gives the derivative of the integral with respect to \(x\).

Key Concepts

Differentiation Under Integral Sign

Differentiating under the integral sign, often called using the Leibniz rule, is a powerful tool in calculus. It simplifies complex problems involving integrals with variable limits.

This method allows you to differentiate an integral with respect to a parameter. This is particularly useful when the upper limit of integration is not constant, but rather a function of that parameter.

• The general form of Leibniz's Rule is:\[ \frac{d}{dx}\int_{a(x)}^{b(x)} f(t, x) \, dt = f(b(x), x)\cdot b'(x) - f(a(x), x)\cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(t, x) \, dt \]

Here,

• \(f(t, x)\) is the integrand function,
• \(a(x)\) and \(b(x)\) are the lower and upper limits, respectively, which can be functions of \(x\).

To apply this technique, simply follow these rules and substitute appropriately. If you encounter constant limits, like seeing \(a(x)\) as a constant, the term \(f(a(x), x)\cdot a'(x)\) disappears as its derivative is zero.

Variable Upper Limit Integration

An integral with a variable upper limit involves a function in the place of an upper limit constant.

For instance, consider an integral of the form \(\int_{0}^{g(x)} f(t) \, dt\). This behaves differently than a standard integral with fixed limits.

In such cases, the upper limit impacts how the integral behaves as \(x\) changes.

• The differentiation process involves computing the derivative of the upper limit function \(g(x)\).
• This affects the result because changing \(x\) influences both \(g(x)\) and the outcome of the integral.

To solve, recognize the derivative of the upper limit, \(g'(x)\), plays a role only at the boundary. For example, in the exercise, with \(y = \int_{0}^{3x} (1 + t^2) \, dt\), the function \(g(x) = 3x\) gives a derivative of \(g'(x) = 3\). Therefore, the derivative of the integral dependent on boundary is modified by this factor, explaining each newly considered part at \(b(x)\).

Partial Derivatives

In the context of Leibniz's rule, partial derivatives are useful when the integrand has an explicit dependency on the parameter you're differentiating with respect to.

Partial derivatives deal with functions of multiple variables, allowing you to focus changes in just one variable while keeping others constant.

• This helps break down complex problems by tackling one variable change at a time.
• Partial derivatives are an integral part of Leibniz's rule when the integrand explicitly includes \(x\).

In our example problem, the function \(1 + t^2\) doesn't explicitly depend on \(x\). Thus, the partial derivative of the integrand with respect to \(x\) is zero.

This means the term \(\int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(t, x) \, dt\) is zero, simplifying our calculations significantly.
The advantage of understanding and leveraging partial derivatives is that they simplify the extension and application of differentiation under specific conditions, ensuring efficiency in solving integral-calculus intersections.