Question

Find $\frac{d}{dx}{\int }_1^{1/x}\frac{e^{xt}}{t}dt.$

Short answer

The derivative is \ \( - \frac{e}{x^3} + \frac{e^x - e}{x} \ \.

Step-by-step solution

Identify the Problem Type

The exercise involves differentiating an integral with variable limits. This is a situation where the Leibniz Rule for differentiating under the integral sign comes into play.

Recall Leibniz Rule

Leibniz rule states that for an integral of the form \ $F(x)={\int }_{a(x)}^{b(x)}f(x,t)dt$, the derivative is given by \ $\frac{d}{dx}F(x)=f(x,b(x))\frac{d}{dx}b(x)-f(x,a(x))\frac{d}{dx}a(x)+{\int }_{a(x)}^{b(x)}\frac{\partial }{\partial x}f(x,t)dt$.

Define the Components

Here, \ $f(x,t)=\frac{e^{xt}}{t}$, \ $a(x)=1$, and \ $b(x)=\frac{1}{x}$.

Differentiate the Limits

Differentiate the limits: \ $\frac{d}{dx}a(x)=0$ and \ $\frac{d}{dx}b(x)=-\frac{1}{x^2}$.

Apply the Leibniz Rule

Substitute all values into the Leibniz Rule: \ $\frac{d}{dx}{\int }_1^{1/x}\frac{e^{xt}}{t}dt=\frac{e^{x(\frac{1}{x})}}{\frac{1}{x}}(-\frac{1}{x^2})-\frac{e^{x\cdot 1}}{1}(0)+{\int }_1^{1/x}\frac{d}{dx}(\frac{e^{xt}}{t})dt$.

Simplify Boundary Terms

The boundary term simplifies to: \ $\frac{e}{x}(-\frac{1}{x^2})-0=-\frac{e}{x^3}$.

Differentiate the Integrand

Now find the partial derivative inside the integral: \ \( \frac{\frac{\partial}{\partial x} (\frac{e^{x t}}{t}) = e^{x t} \frac{\partial x t}{\frac{\partial x}} = \frac{e^{x t}}{1} = t e^{x t} \ \).

Integrate the New Expression

Thus, the integral of the partial derivative is: \ ${\int }_1^{1/x}x{e}^{xt}dt$. Perform the integration: \ $\frac{e^x-e}{x}$.

Put It All Together

Summing all contributions: \ $-\frac{e}{x^3}+\frac{e^x-e}{x}$.

Key concepts

Variable Limits

In calculus, integrals with variable limits are essential for modeling scenarios where the extent of integration changes. When limits are functions of the differentiation variable, it requires special techniques to handle. This problem involves navigating these variable limits using the Leibniz Rule.

To understand how variable limits impact differentiation, consider the setup: you integrate a function where the upper or lower limits are not constants, but functions of the variable of interest. For example, given an integral from a(x) to b(x), these limits vary as x changes.

In our example, the integral $\frac{d}{dx}{\int }_1^{1/x}\frac{e^{xt}}{t}\text{d}t$ has the limit functions a(x) = 1 and b(x) = \frac{1}{x}\. Thus, as x varies, the range of integration shifts from a fixed point to another dependent on x.

Differentiation Under the Integral Sign

Differentiating under the integral sign, also known as the Leibniz rule, is a powerful technique in calculus. It allows us to differentiate an integral whose limits or integrand, or both, depend on the differentiation variable. This technique retains the process of differentiation and ensures seamless handling of variable limits.

The essence of the Leibniz rule is described by the formula:

$$F(x)={\int }_{a(x)}^{b(x)}f(x,t)\text{d}t$$

\raisebox{-1pt}{\textrm{The Leibniz derivative formula}}:
$$\frac{d}{dx}F(x)=f(x,b(x))\frac{d}{dx}b(x)-f(x,a(x))\frac{d}{dx}a(x)+{\int }_{a(x)}^{b(x)}\frac{\text{raisebox}-1ptpartialderivative}{\text{raisebox}-1ptpartialx}f(x,t)\text{d}t$$
Note that this does not directly differentiate through the integrand but includes border terms and the partial derivative of the integrand, ensuring complete differentiation under the integral sign.

Partial Derivative

When dealing with functions dependent on multiple variables, partial derivatives come into play. They measure the rate of change with respect to one variable while keeping others constant. In differentiating under the integral sign, we often find the partial derivative of the integrand concerning the differentiation variable.

Here, in the integral $\frac{d}{dx}{\int }_1^{1/x}\frac{e^{xt}}{t}\text{d}t$, our integrand is $\frac{e^{xt}}{t}$. The partial derivative of this integrand concerning x is:

$$\frac{\text{raisebox}-1ptpartial}{\text{raisebox}-1ptpartialx}\frac{e^{xt}}{t}=\frac{t{e}^{xt}}{t}=t{e}^{xt}\boxed{{\displaystyle }}$$
This result gives us an expression that we will eventually integrate back over the same variable limits.

Integration Techniques

Applying the Leibniz rule and partial derivatives, we reach the point of actual integration. Integration techniques involve simplifying and evaluating the integral of the given expression.

For the partial derivative derived previously, $t{e}^{xt}$, we need to integrate this over the variable limits from \ 1 \ to \ \frac{1}{x} \:

$$\frac{d}{dx}{\int }_1^{1/x}\frac{e^{xt}}{t}\text{d}t=-\frac{e}{x^3}+{\int }_1^{1/x}t{e}^{xt}\text{d}t$$
To solve \ \int_{1}^{1/x} t e^{xt} \text{d}t \, let's perform the integral:

$${\int }_1^{1/x}t{e}^{xt}\text{d}t=\frac{e^{xx}-e}{x}=\frac{e^x-e}{x}$$
Adding this to the boundary term \ -\frac{e}{x^3}\boxed{}{ gives us the result... Combining all, we get:

$$-\frac{e}{x^3}+\frac{e^x-e}{x}$$
This showcases the intricate application of integration techniques combined with differentiation under the integral sign. By consolidating these skills, one solves complex, variable integral problems efficiently.