### Video Transcript

The output in milligrams of a chemical reaction after 𝑡 seconds is given by 𝑦 equals four 𝑡 cubed. What is the rate of production of this reaction at 𝑡 equals two seconds?

The rate of change of any function at time 𝑡 is given by d𝑦 by d𝑡. This means that we need to differentiate the equation with respect to 𝑡. If the function 𝑦 is equal to 𝑎 multiplied by 𝑡 to the power of 𝑛, then d𝑦 by d𝑡 is equal to 𝑛 multiplied by 𝑎 multiplied by 𝑡 to the power of 𝑛 minus one. We multiply the power by the coefficient and reduce the power by one.

In this question, we were told that 𝑦 is equal to four 𝑡 cubed. In order to work out d𝑦 by d𝑡, we firstly need to multiply the power by the coefficient. Three multiplied by four is equal to 12. The power reduces by one. Therefore, d𝑦 by d𝑡 is equal to 12𝑡 squared. We were asked to calculate the rate of production at 𝑡 equals two. Therefore, we need to substitute this into the equation. d𝑦 by d𝑡 at time 𝑡 equals two is equal to 12 multiplied by two squared. Two squared is equal to four. And 12 multiplied by four is equal to 48.

If the output of a chemical reaction is given by the formula 𝑦 equals four 𝑡 cubed. Then, the rate of production at 𝑡 equals two seconds is 48 milligrams per second.