Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(G^{\prime}(x)\). $$ G(x)=\int_{-x^{2}}^{x} \frac{t^{2}}{1+t^{2}} d t $$

Step-by-Step Solution

Answer

\(G'(x) = \frac{x^2}{1+x^2} + \frac{2x^5}{1+x^4}\).

Step 1: Understand the Given Function

We are given a function \(G(x) = \int_{-x^2}^{x} \frac{t^2}{1+t^2} \, dt\). Our task is to find \(G'(x)\), which is the derivative of this function with respect to \(x\). The function \(G(x)\) is a definite integral with variable limits of integration.

Step 2: Apply the Leibniz Rule

To find the derivative of a definite integral with variable limits, we use the Leibniz rule which states: if \(G(x) = \int_{a(x)}^{b(x)} f(t) \, dt\), then \(G'(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)\). In our case, \(a(x) = -x^2\) and \(b(x) = x\).

Step 3: Differentiate the Limits

Calculate the derivatives of the limits: \(a'(x)\) and \(b'(x)\). For the upper limit, \(b(x) = x\), so \(b'(x) = 1\). For the lower limit, \(a(x) = -x^2\), so \(a'(x) = -2x\).

Step 4: Evaluate the Function at the Limits

Evaluate the integrand \(f(t) = \frac{t^2}{1+t^2}\) at the limits \(t = b(x) = x\) and \(t = a(x) = -x^2\). Thus, \(f(b(x)) = \frac{x^2}{1+x^2}\) and \(f(a(x)) = \frac{(-x^2)^2}{1+(-x^2)^2} = \frac{x^4}{1+x^4}\).

Step 5: Assemble the Derivative Using the Leibniz Rule

Substitute the evaluated values and their respective derived steps into the Leibniz formula: \(G'(x) = \left(\frac{x^2}{1+x^2}\right)(1) - \left(\frac{x^4}{1+x^4}\right)(-2x)\).

Step 6: Simplify the Expression

Simplify the expression: \(G'(x) = \frac{x^2}{1+x^2} + \frac{2x \cdot x^4}{1+x^4}\) becomes \(G'(x) = \frac{x^2}{1+x^2} + \frac{2x^5}{1+x^4}\).

Key Concepts

Leibniz Rule

The Leibniz Rule is a powerful tool in calculus used to find the derivative of an integral whose limits are functions of a variable, usually in the form \( x \).
This rule is particularly useful when dealing with definite integrals in which the limits of integration are not constant.
It states that for a function \( G(x) = \int_{a(x)}^{b(x)} f(t) \, dt \), the derivative \( G'(x) \) can be found using:

• \( G'(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x) \)

Here, \( f(t) \) is the integrand, \( a(x) \) and \( b(x) \) are the variable limits of the integral, and \( a'(x) \) and \( b'(x) \) are their respective derivatives.
This rule leverages both the evaluation of the function at the boundary limits and their respective rates of change, providing insights into how the function \( G(x) \) behaves as \( x \) changes.

Definite Integral

A definite integral is a fundamental concept in calculus that calculates the net "accumulation" of quantities, often represented as the area under a curve, between two boundary points or limits.
The notation \( \int_{a}^{b} f(t) \, dt \) denotes integrating the function \( f(t) \) from \( t = a \) to \( t = b \).

• The lower limit \( a \) and the upper limit \( b \) are usually constants, providing a fixed range over which the function is evaluated.
• When the limits are variables, the integral is said to have variable limits, introducing the potential for more complex behavior and requiring additional rules, like the Leibniz Rule, to differentiate it.

Therefore, the definite integral gives the total quantity, accounting for any positive or negative contributions from the function, within the range of integration.
In the given problem, \( G(x) \) being a definite integral, means it evaluates the function \( f(t) = \frac{t^2}{1+t^2} \) over the variable limits \( -x^2 \) to \( x \).

Derivative of Function with Variable Limits of Integration

When finding the derivative of a function defined by an integral with variable limits, it is not just enough to follow basic differentiation rules.
We must account for both the behavior of the function within those limits and how those limits themselves change.
The derivative \( G'(x) \) of such a function, \( G(x) = \int_{a(x)}^{b(x)} f(t) \, dt \), involves:

• Evaluating the function \( f(t) \) at the upper and lower variable limits.
• Computing the derivatives \( a'(x) \) and \( b'(x) \) of these limits with respect to \( x \).

The formulation involves substituting these results into the equation derived from the Leibniz Rule.
The example provided illustrates these steps: calculating \( f(-x^2) \), \( f(x) \), \( a'(x) \), \( b'(x) \), and finally the derivative \( G'(x) \) by combining these evaluations appropriately.
This approach allows the understanding of how changes in \( x \) influence \( G(x) \), giving insight into the overall behavior of this integral-based function.