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Consider the following profit function.
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We have p of q equal to 76 q times the exponential of minus 0 .5 q, where q here ranges 0 and 100.
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And q represents the number of units produced.
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In question a, we want to find the rate of change of the profit function.
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We need that.
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We want to find p prime of q.
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Let's evaluate our first derivative of the following function.
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So here we see that we're going to need to employ product rule, because we have the product of two functions.
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Let's establish 76 q as our first function and the exponential of our second function.
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And product rule states that the derivative, our product of functions will give us the derivative of our first function, 76 q times our second function, the exponential here.
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Plus our first function times the derivative of our second function.
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So now let's evaluate the derivative of 76q.
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This will simply give us 76.
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The derivative of our exponential will return the same exponential, from which we will need to multiply by the derivative of the exponent here, minus 0 .5q...