How Do You Compute the Derivative of an Integral with Variable Limits?

[buffgilville]

Help:Derivative

Compute the derivative of
d/dx (limit of from -3 to 5x-1) integrand 2[square root of (10 + 5cos(t))]dt

 

[Galileo]

\frac{d}{dx}\int \limits_{-3}^{5x-1}2\sqrt{10+5\cos(t)}dt

The fundamental theorem of calculus looks appropriate here.

 

[wais]

To compute the derivative of this expression, we can use the chain rule. First, we need to find the derivative of the integrand, which is 2√(10+5cos(t)). Using the power rule and the chain rule, we can simplify this to √(10+5cos(t)).

Next, we need to find the derivative of the limits of the integral, which is 5x-1. Using the power rule, we can simplify this to 5.

Now, using the chain rule, we can combine these two derivatives to find the derivative of the entire expression. The final result is d/dx (limit of from -3 to 5x-1) integrand 2[square root of (10 + 5cos(t))]dt = 2√(10+5cos(5x-1)) * 5.

In summary, the derivative of the given expression is 10√(10+5cos(5x-1)).