Question

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{-2}^{x} \sqrt{1+e^{5 t}} d t$$

Short answer

The derivative of the given function is \(dy/dx = \sqrt{1+e^{5x}}\)

Step-by-step solution

- Identify the integrand

The function inside the integral, \( \sqrt{1+e^{5t}} \), is the integrand. This function needs to be evaluated at the upper limit of the integral once we apply the Fundamental Theorem of Calculus.

- Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that the derivative of an integral with respect to its upper limit is just the integrand evaluated at that limit. In this case, the upper limit of the integral is \(x\), so the derivative of y with respect to x, \(dy/dx\), is \( \sqrt{1+e^{5x}} \).

- Write the Final Answer

With the application of the Fundamental Theorem of Calculus, we have found that \(dy/dx = \sqrt{1+e^{5x}}\). This is the final answer.