Quotient rule calculus6/22/2023 ![]() ![]() It is one of the basic, simple and widely used rule to differentiate equations. But if you don't know the chain rule yet, this is fairly useful. The quotient rule is a fundamental rule in differential calculus. Quotient Rule Calculus Tutorials Quotient Rule Suppose we are working with a function h ( x) that is a ratio of two functions f ( x) and g ( x). But you could also do the quotient rule using the product and the chain rule that you might learn in the future. Now what you'll see in the future you might already know something called the chain rule, or you might You could try to simplify it, in fact, there's not an obvious way ![]() Plus, X squared X squared times sine of X. This is going to be equal to let's see, we're gonna get two X times cosine of X. Combine the differentiation rules to find the derivative of a polynomial or rational function. Extend the power rule to functions with negative exponents. Use the quotient rule for finding the derivative of a quotient of functions. Actually, let me write it like that just to make it a little bit clearer. Use the product rule for finding the derivative of a product of functions. So that's cosine of X and I'm going to square it. All of that over all of that over the denominator function squared. The derivative of cosine of X is negative sine X. Minus the numerator function which is just X squared. V of X is just cosine of X times cosine of X. So it's gonna be two X times the denominator function. You can use the product rule to differentiate Q (x), and the 1/ (g (x)) can be differentiated using chain rule with u g (x), and 1/ (g (x)) 1/u. So based on that F prime of X is going to be equal to the derivative of the numerator function that's two X, right over The quotient rule could be seen as an application of the product and chain rules. Of X with respect to X is equal to negative sine of X. So that is U of X and U prime of X would be equal to two X. Well what could be our U of X and what could be our V of X? Well, our U of X could be our X squared. But it is simpler to do this: d dx 10 x2 d dx10x 2 20x 3. If we do use it here, we get d dx10 x2 x2 0 10 2x x4 20 x3, since the derivative of 10 is 0. (F) can be found by applying the quotient rule and then using the sum and constant multiple rules to differentiate the numerator and the product rule to differentiate the denominator. Of course you can use the quotient rule, but it is usually not the easiest method. So let's say that we have F of X is equal to X squared over cosine of X. In calculus notation, the latter two facts tell us that (N(100). We would then divide by the denominator function squared. Get if we took the derivative this was a plus sign. If this was U of X times V of X then this is what we would The denominator function times V prime of X. Its going to be equal to the derivative of the numerator function. Then the quotient rule tells us that F prime of X is going to be equal to and this is going to lookĪ little bit complicated but once we apply it, you'll hopefully get a little bit more comfortable with it. So for example if I have some function F of X and it can be expressed as the quotient of two expressions. But here, we'll learn about what it is and how and where to actually apply it. The quotient rule can be used to differentiate the tangent function tan (x), because of a basic identity, taken from trigonometry: tan (x) sin (x) / cos (x). It using the product rule and we'll see it has some Note that we first use linearity of the derivative to pull the 10 out in front.Going to do in this video is introduce ourselves to the quotient rule.
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