Question

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{x}^{7} \sqrt{2 t^{4}+t+1} d t$$

Short answer

The derivative of the given function is \(dy/dx = -(\sqrt{2 x^{4}+x+1})\).

Step-by-step solution

Identify the integral function

The function given by the problem to find the derivative of is \(y=\int_{x}^{7} \sqrt{2 t^{4}+t+1} dt\). This is an integral with moving boundaries.

Swap the limits of Integration

Since the Fundamental Theorem of Calculus applies for the integral from a constant to x, and in our case the roles of x and 7 are reversed, it's needed to swap the limits of integration. This will change the sign of the integral. Thus, \(y = -\int_{7}^{x} \sqrt{2 t^{4}+t+1} dt\).

Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, the derivative of the given function is the function inside the integral evaluated at x. So, \(dy/dx = -(\sqrt{2 x^{4}+x+1})\).