Differentiation

Take log: At , , so:

Given: Differentiate both sides w.r.t. x: But we also know: Eventually, you'll get:

We are given that the rate at which the volume of the balloon changes is: The volume of a sphere is given by: To find the rate at which the radius changes, differentiate the volume with respect to time: Substitute the values: Solve for :

We are given the equation . To differentiate implicitly, we apply logarithms to both sides. However, based on the structure of the equation, solving for explicitly may lead to inconsistencies or undefined solutions in this case, implying that no valid differentiation exists under the given conditions.

Step 1: Differentiate the function.

Step 2: Substitute .

Step 3: Simplify.

Step 4: Correct evaluation.