Video Transcript
The work done by an engine at time 𝑡 is given by the relation 𝑊 of 𝑡 is equal to one over five plus 𝑡 joules. Find the power of the engine as a function of time.
We begin by recalling that power as a function of time 𝑃 of 𝑡 is equal to d by d𝑡 of 𝑊 of 𝑡. In other words, to find an expression for the power, we differentiate our expression for the work done. In this question, 𝑊 of 𝑡 is equal to one over five plus 𝑡, which can be rewritten as five plus 𝑡 to the power of negative one. One way of differentiating this expression would be using the chain rule, where we could let 𝑢 equal five plus 𝑡. Alternatively, we could use the general rule that if 𝑦 is equal to 𝑎 plus 𝑏𝑥 to the power of 𝑛, then d𝑦 by d𝑥 is equal to 𝑏 multiplied by 𝑛 multiplied by 𝑎 plus 𝑏𝑥 to the power of 𝑛 minus one.
In our expression, the power 𝑛 is negative one, our value of 𝑎 is five, and 𝑏 is equal to one. It is the coefficient of 𝑡. As our variable is 𝑡, differentiating with respect to 𝑡 gives us one multiplied by negative one multiplied by five plus 𝑡 to the power of negative two. Using our laws of exponents or indices, this can be rewritten as negative one over five plus 𝑡 all squared. Since the work done was measured in the standard units of joules, our power will be measured in watts. And we can therefore conclude that the power of the engine as a function of time is negative one over five plus 𝑡 all squared watts.
