### Video Transcript

The work done by an engine at time 𝑡 is given by the relation 𝑤 of 𝑡 is equal to negative 12 multiplied by the cos of two 𝑡 joules. Find the power of the engine as a function of time.

In this question, we are given an expression for the work done by an engine in terms of time 𝑡. This is equal to negative 12 multiplied by cos two 𝑡, where the work is measured in the standard units of joules. We know that the power supplied by a force is the time derivative of the work done. This means that we can find an expression for 𝑃 of 𝑡 by differentiating our expression for the work done.

When differentiating the cosine function, if 𝑦 is equal to 𝑎 multiplied by the cos of 𝑛𝑥, then d𝑦 by d𝑥 is equal to negative 𝑎𝑛 multiplied by the sin of 𝑛𝑥. This means that differentiating our function with respect to 𝑡 gives us negative negative 12 multiplied by two multiplied by sin of two 𝑡, and this simplifies to 24 multiplied by sin of two 𝑡. As we are working in standard units, the expression for power will be in terms of watts.

The power of the engine as a function of time is equal to 24 multiplied by sin of two 𝑡 watts.