Question

For the following exercises, evaluate by any method. $$ \frac{d}{d x} \int_{x}^{x^{2}} \frac{d t}{t} $$

Short answer

The derivative is \( \frac{1}{x} \).

Step-by-step solution

Recognize the Function

The problem is to find the derivative of an integral, which can be approached using the Leibniz rule for differentiating under the integral sign. The expression is \( \frac{d}{dx} \int_{x}^{x^2} \frac{dt}{t} \).

Identify the Leibniz Rule

The Leibniz rule states: \[ \frac{d}{dx} \int_{a(x)}^{b(x)} f(t,x) \, dt = f(b(x),x)b'(x) - f(a(x),x)a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x}f(t,x) \, dt\]For our problem, \( f(t) = \frac{1}{t} \) and thus does not directly depend on \( x \), simplifying the partial derivative term.

Apply the Leibniz Rule

Substitute into the Leibniz rule: Let \( f(t)=\frac{1}{t} \), thus both \( a(x) = x \), \( b(x) = x^2 \).Then, \( f(b(x)) = \frac{1}{x^2} \), \( f(a(x)) = \frac{1}{x} \), \( b'(x) = 2x \), and \( a'(x) = 1 \).The expression becomes: \[ \frac{1}{x^2}(2x) - \frac{1}{x}(1) \]

Simplifying the Expression

Simplify the expression: \[ \frac{2x}{x^2} - \frac{1}{x} = \frac{2}{x} - \frac{1}{x} \]Further simplify: \[ \frac{1}{x} \]

Key concepts

Leibniz Rule

The Leibniz Rule is a powerful tool in calculus for taking the derivative of an integral whose limits are functions of the variable of differentiation. It allows us to differentiate integrals where either the upper or lower limit, or both, depend on the variable. The rule can be stated in the following form: \[ \frac{d}{dx} \int_{a(x)}^{b(x)} f(t, x) \, dt = f(b(x), x)b'(x) - f(a(x), x)a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x}f(t, x) \, dt \] This formula requires:

• \( a(x) \) and \( b(x) \): the lower and upper limits of the integral, both functions of \( x \).
• \( f(t, x) \): the function being integrated.
• \( b'(x) \) and \( a'(x) \): derivatives of the upper and lower limits with respect to \( x \).

This rule becomes especially useful when analyzing complex functions in physics and engineering applications. In our problem, understanding the role of the Leibniz Rule helps us evaluate the integral smoothly.

Differentiation under the Integral Sign

Differentiation under the integral sign is a method that allows us to differentiate an integral with respect to a parameter, usually when the limits of integration are not constant. This technique can handle cases where the limits and/or the integrand is a function of the differentiation variable.
The process involves replacing the function inside the integral with one that includes the parameter of interest, and then computing the derivative outside the integral. For example, we use the Leibniz Rule to manage problems like \[ \frac{d}{dx} \int_{x}^{x^2} \frac{dt}{t} \] which associates with changing limits. By leveraging the partial derivative aspect of the Leibniz Rule, any direct dependence of the integrand on \( x \) can be addressed.

Evaluation of Integrals

Evaluating integrals is a central task in calculus. It involves finding the function integral associated with a given function over a specified domain. A solid grasp of methods for calculating definite and indefinite integrals is crucial. Integrals allow us to calculate areas, volumes, central points, and many other useful things.
In our specific exercise, we are evaluating an integral where both limits are functions of \( x \). Standard techniques, like substitution and parts, may not be directly applicable. The Leibniz Rule, again, offers a direct approach to differentiate such integrals. It effectively transforms a complex integral into operations involving basic differentiation and evaluation at the limits.

• Recognize function forms and applicability of known rules.
• Understand implications of changing integration limits.

Derivative of an Integral

The derivative of an integral addresses situations where we need to differentiate a function defined as an integral of another function. Generally, when the limits of an integral depend on the differentiating variable, the exercise requires more sophisticated techniques than simple fundamental theorem applications.
For example, considering \[ \frac{d}{dx} \int_{x}^{x^2} \frac{dt}{t} \] , we apply the Leibniz Rule because the limits \( x \) and \( x^2 \) are variable. By substituting into the Leibniz Rule formula, evaluating derivatives of the limits, and simplifying, we eventually determine that the 'net' derivative simplifies to \( \frac{1}{x} \).
Operations such as these highlight the interplay between derivative rules and integral calculus, enriching the computational landscape for students.