Let’s look at another example to make sure you got the basics down. Usually the first shortcut rule you study for finding derivatives is the power rule.

Find: \(\displaystyle\int \dfrac{3}{x^5} – \dfrac{1}{4x^2} \text{ dx}\). In terms of ln(x), these state: Using these, you can expand an expression before trying to find the derivative, as you can see in the next few examples. If \(C\left( x \right)\) is the cost function for some item then the average cost function is.

Since this is not simply \(\ln(x)\), we cannot apply the basic rule for the derivative of the natural log.

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(and NO I won’t do your math homework for you). Finally, the marginal revenue function is \(R'\left( x \right)\) and the marginal profit function is \(P'\left( x \right)\) and these represent the revenue and profit respectively if one more unit is sold. But, if we combine this with the laws of logarithms we can do even more. If you really want to get better at calculus, following these problems is a great way to make yourself practice! Solution b: While we can do this example by hand, we also want to use it to set up a solution with Excel, since we may want help on problems where the numbers are not as nice. For each of these, you can simply apply the power rule without any algebra at all.

Either of the last two lines can be used as a final answer, but the last one looks a little nicer and is probably going to be preferred by your teacher if you are currently taking calculus! \(\begin{align} y^{\prime} &= \left(2x^4 – 5x^2 + 1\right)^{\prime}\\ &= \left(2x^4\right)^{\prime} – \left(5x^2\right)^{\prime} + \left(1\right)^{\prime}\end{align}\), \(= 2\left(x^4\right)^{\prime} – 5\left(x^2\right)^{\prime} + \left(1\right)^{\prime}\). Therefore, we can write the final answer as: You may think this is all you can really do with the power rule. Identify the objective function. The result is an example of a differential equation. One way of thinking about the derivative, is as the slope of a function at a given point. If you are willing to put in a little bit of outside effort, many topics in trig are easy to pick up and there are really only a few key skills/ideas. If they sell x widgets during the year then their profit, in dollars, is given by, What is the marginal cost when \(x = 175\) and \(x = 300\)? First, let’s look at the more obvious cases.

More simply, you can think of the quotient rule as applying to functions that are written out as fractions, where the numerator and the denominator are both themselves functions.

Since this cannot be simplified, we have our final answer. Let’s now turn our attention to the average cost function. However, this average cost function is fairly typical for average cost functions so let’s instead differentiate the general formula above using the quotient rule and see what we have. will be, \(y = \dfrac{2}{x^4} – \dfrac{1}{x^2}\). And by having access to our ebooks online or by storing it on your computer, you have convenient answers with Business Calculus Problems Answers .

Normally, if it was just \(\ln(x)\), you would say the derivative is \(\dfrac{1}{x}\). Using this rule, we can take a function written with a root and find its derivative using the power rule. Remember that this is just algebra – no calculus is involved just yet. This function is the product of two simpler functions: \(x^4\) and \(\ln(x)\).

With a little bit of practice, you will probably be able to write the derivative of this function down without thinking. This week’s problem: In this part all we need to do is get the derivative and then compute \(C'\left( {300} \right)\). You may be wondering what happened to \(\ln(5)\).

You are standing some distance away from it. \(2\displaystyle\int x^3\text{ dx} + 4\displaystyle\int x^2 \text{ dx} = 2\left(\dfrac{x^{3+1}}{3+1}\right) + 4\left(\dfrac{x^{2+1}}{2+1}\right) + C\). Let’s look at one more example without so much explanation to distract us.



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