23–51. Calculating derivatives Find the derivative of the foll...

Question

23–51. Calculating derivatives Find the derivative of the following functions.

y = a sin x + b cos x/a sin x - b cos x; a and b are nonzero constants

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of the function F X equals 2 + X + 3 X X divided by 2 + X minus 3 X. A says it's 12 divided by 2 + X + 3 X. B, 12 divided by 2 + X minus 3 X. C, 12 divided by 2 X + 3 X squared, and D, 12 divided by 2 + X minus 3 X X squared. Now, how can we find the derivative of FF X which is a cautient? What do we know about the derivatives of quotient? Well, recall we can use the quotient rule. The quotient rule basically tells us that if we have a function Y equals u divided by V, where U and V are functions of X, then the derivative of Y is going to be equal to U U V, OK. Minusuv divided by v2 where U and V are the derivatives of U and V respectively. So in other words, if we can differentiate our numerator U and our denominator V, then we should be able to find the derivative of the function by the quotient rule. So let's go ahead and differentiate. First, starting with our numerator, we know that U equals 2 x plus 3 x x. Now if we differentiate uh this this term here, if they differentiate this expression. Then the derivative of 2 + X is negative to sin X because the derivative of X is negative sin X and the derivative of 3 sin X is going to be 3 X. The derivative of the sine function is the cosine. Next, we know that our denominator V is 2 + X minus 3 X. And when we differentiate V. We're going to get negative 2 sin X again based on the same definition, minus 3+ X. Now that we have UV, and their derivatives, we can apply the quotient rule. Because now, this tells us that the derivative of F of X will be equal to U multiplied by V. OK. So that's gonna be negative to sin x. Plus 3 X multiplied by 2 X. -3 X, OK. Minus UV, OK? So now, that's going to be 2 + X + 3 X. OK. Multiplied by -2 X X minus 3 + X. No, all of this is gonna be divided by V squared. So that's 2 cost X. Minus 3 x x squared. No, where do we go from here? Well, let's try to simplify our numerator, OK? And we can simplify it by multiplying our linear expressions using the foil method. We're going to multiply the outside, inside, outside, the first, outside, inside, and the last terms, OK? And then, after we do that, for the second product, we'll factor out the negative sign. So no, when we go ahead here. Let me do that, let me make some space down here. Then, for our first product, we're going to have negative 4 X X X, OK, plus 6 square x, OK, plus oops, plus 6 square x, minus 9 X X plus X. Now for our second product here, OK, then we're gonna have 4 negative, 4 sin x, X, OK. -4 X X X minus 6 squared X. OK. minus 6 square x, OK. minus 9 x x cost X. OK. I hope you can, I hope we have enough space here. And again, all of this is being divided by 2 X minus 3 X X squared. Now we can distribute our factor out to our negative sign, OK? Let me bring this over so that you can uh see everything. Now we can factor out our negative sign because no. If we subtract, well, all of these terms are negative, so when you factor are negative sign, they'll be positive. So now this, let me put this in, in black here. OK, so this is gonna be plus 4 x + X plus 6 X 6 squared X plus 6 square x plus 9 s x + X. And no, if we go ahead and simplify here, -9 sin x x + 9 X X is 0, -4 sin X X plus 4 x + X is 0. And no, we're left with our sin squared X and cost squared x terms, which when we simplify, that's gonna be 6 sin squared X plus 6 squared X 6 squared X plus 6 square x 12 sin square x. Plus 6 squared X plus 6 squared X, which is 12 squared X. All of this is still divided by our denominator. Now there are two things we noticed there. First of all, we can factor 012 to get 12 multiplied by a sin squared X plus squared X divided by our denominator. Now another thing we notice here is that in our numerator, we have a a trigonometric identity. OK, recall. Recall, OK, that sin square x plus square x equals 1, OK. That's the fundamental trigonometric identity. So with that idea here, since this is 1, then that means our derivative then of F of X is going to be 12 divided by 2 class X minus 3 sin X. Squared Therefore, when we look back at our answer twice, that tells us D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

23–51. Calculating derivatives Find the derivative of the following functions.
y = a sin x + b cos x/a sin x - b cos x; a and b are nonzero constants

Key Concepts

Derivatives

Trigonometric Functions

Quotient Rule

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