Question

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{x}^{0} \ln \left(1+t^{2}\right) d t$$

Short answer

The derivative of the function \(y\) with respect to \(x\), or \(dy/dx\), is \(-\ln (1+x^{2})\).

Step-by-step solution

Understand the Problem

Looking at the given function \(y=\int_{x}^{0} \ln \left(1+t^{2}\right) d t\), it is clear that this is an example of a function defined by an integral. Here, the limits of integration are \(x\) and \(0\) and the function to be integrated is \(\ln \left(1+t^{2}\right)\). The goal in this exercise is to find the derivative of \(y\) with respect to \(x\), or \(dy/dx\). The Fundamental Theorem of Calculus will provide a helpful start.

Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that the derivative of an integral of a function is simply the original function. Therefore, if we take the derivative of both sides, we get \(dy/dx\). But since the lower limit of our integral is \(x\) and upper limit is a constant, we have to account for this by negating our result. So, \(\frac{dy}{dx} = - \ln (1+x^{2})\).

Final answer

The derivative of the function \(y\) with respect to \(x\), \(dy/dx\), is given by \(- \ln (1+x^{2})\).