### Video Transcript

A populationβs size after π‘ days is given by π of π‘ is equal to 11π‘ squared plus 35,923. Find the instantaneous rate of change in the population when π‘ equals 12.

In this question, we are asked to find the instantaneous rate of change in the population. And we are asked to calculate this when π‘ is equal to 12. This means the derivative of the population function π of π‘ with respect to π‘ when π‘ is 12.

We know by definition that π prime of π is equal to the limit as β approaches zero of π of π plus β minus π of π all divided by β. In this question, we need to calculate π prime of 12 as shown. We will begin by using our expression for π of π‘ to find an expression for π of 12 plus β. Replacing π‘ with 12 plus β, we have 11 multiplied by 12 plus β squared plus 35,923. Expanding 12 plus β squared gives us 144 plus 24β plus β squared. And multiplying through by 11, we have π of 12 plus β is equal to 1,584 plus 264β plus 11β squared plus 35,923.

Next, we can find an expression for π of 12. This is equal to 11 multiplied by 12 squared plus 35,923. And 11 multiplied by 12 squared is equal to 1,584. We can now substitute these into our expression for π prime of 12. The two constants on the numerator will cancel. And we are left with the limit as β approaches zero of 264β plus 11β squared all divided by β. Since β is approaching zero and will therefore never be equal to zero, we can divide the numerator and denominator by β. This gives us the limit as β approaches zero of 264 plus 11β.

As we now have a polynomial in terms of β, we can use direct substitution such that π prime of 12 is equal to 264. And we can therefore conclude that the instantaneous rate of change in the population when π‘ is equal to 12 is 264.