[1/1 Points] DETAILS MY NOTES HARMATHAP12 9.4.050. The monthly output of a certain product is \[ Q(x)=1500 x^{5 / 2} \] where \( x \) is the capital investment in millions of dollars. Find \( d Q / d x \), which can be used to estimate the effect on the output if an additional capital investment of \( \$ 1 \) million is made. \( d Q / d x= \) 2. [1/1 Points] DETAILS MY NOTES

To find \( \frac{dQ}{dx} \), we need to differentiate the given function \( Q(x) = 1500 x^{5/2} \). Using the power rule for differentiation, we get: \[ \frac{dQ}{dx} = 1500 \cdot \frac{5}{2} x^{(5/2)-1} = 1500 \cdot \frac{5}{2} x^{3/2} = 3750 x^{3/2} \] Now, you can substitute the specific capital investment value for \( x \) to estimate the output effect for that additional million dollars. Consider capital investments leading to growth spurts. In the early 20th century, many industries saw dramatic increases in output due to capital influxes for machinery and infrastructure. For example, the automotive industry boomed after significant investments in assembly lines, showcasing how a few million dollars could accelerate production output dramatically—much like the mathematical relationship you’re exploring! When it comes to estimating the impact of an additional dollar on output, a common pitfall is neglecting to evaluate the function at relevant values of \( x \). Always remember to use \( x \) corresponding to the current level of investment. If you take the derivative without substituting a practical \( x \), you might get numbers without context, hence miss a chance to strategize your investment decisions effectively!