Suppose that a company’s revenue of producing 𝑥 units of a certain item is given by the equation 𝑅 of 𝑥 equals 3000 plus 7.4𝑥 plus 0.75𝑥 squared. What is the instantaneous rate of change of 𝑅 with respect to 𝑥 when 𝑥 equals 200?
The instantaneous rate of change of the quantity is its rate of change at a given moment in time. And it can be found by evaluating the first derivative of that function at the given moment. We therefore, need to evaluate 𝑅 prime of 𝑥 or d𝑅 by d𝑥 as we’re looking for the rate of change of 𝑅 with respect to 𝑥 when 𝑥 is equal to 200. We can find the first derivative of our function 𝑅 of 𝑥 with respect to 𝑥 by using the power rule of differentiation. First, we need to differentiate 3000 and the derivative of a constant with respect to 𝑥 is just zero.
Next, we need to differentiate positive 7.4𝑥. And its derivative is just 7.4. It’s actually 7.4 multiplied by 𝑥 to the power of zero. But as 𝑥 to the power of zero is just one, this simplifies to 7.4. Finally, we differentiate the third term, positive 0.75𝑥 squared. And this derivative is equal to positive 0.75 multiplied by two 𝑥. We bring down the power of two. And then we decrease the power by one. Two multiplied by 0.75 is 1.5. So our first derivative simplifies to 𝑅 prime of 𝑥 is equal to 7.4 plus 1.5𝑥.
Finally, we evaluate this derivative at the given moment. That’s when 𝑥 is equal to 200, giving 𝑅 prime of 200 equals 7.4 plus 1.5 multiplied by 200. 1.5 multiplied by 200 can be thought of as three over two multiplied by 200. And then cancelling a factor of two, we have three over one multiplied by 100 which is 300. So 𝑅 prime of 200 is equal to 7.4 plus 300; that’s 307.4. So by using differentiation to find the first derivative of our function 𝑅 of 𝑥 and then evaluating this first derivative 𝑅 prime of 𝑥 when 𝑥 is equal to 200, we see that the instantaneous rate of change of 𝑅 with respect to 𝑥 when 𝑥 equals 200 is 307.4.