Find the derivative of the following functions.
y = cot ...

Question

Find the derivative of the following functions.

y = cot x / (1 + csc x)

Accepted Answer

Welcome back, everyone. In this problem, we want to find the derivative of Y E equals 10 X divided by 3 + 6 X. Now notice that Y is a caution. What do we know about finding the derivatives of cautions? Well, recall, OK, that if we have a function Y. That's equal to u divided by V where U and V are differentiable functions. Then the derivative of Y equals Vu minusuv divided by V2. OK, where U and V are the derivatives of U and V respectively. If we look at our function Y, U, the numerator is 10 X and V, the denominator is 3 + 6 X. So if we can differentiate both of those functions with respect to X, then we should be able to find the derivative of Y by the quotient rule. So let's go ahead and do that, OK? Now, for our function, given that you, let me write it here, U is equal to the tangent of X, OK. Then if we differentiate 1 X, then U. OK. It is going to be equal to U is gonna be equal to. 6 squared X, that's the derivative of the tangent of X. Let me bring it down here just to make some space, OK. Likewise, if we take V, which is equal to 3 + X, OK, then. The derivative of V is going to be equal to X1 X. The derivative of 3 is 0 and the derivative of sec X is se X1 X. Now that we have U, V, and its derivatives, we can go ahead and apply the quotient rule. Because no, this tells us then. That the derivative of Y will be equal to V multiplied by U. So that's 3 + 6 X. Multiplied by 6 squared X. Minus you multiplied by V. OK, so that's gonna be 10 X. Multiplied by sec X1 X. I know all of this is being divided by the squared. 3 + X2. Now, let's try to simplify our expression. We, we have the derivative, but we want to write it in a simpler way. Now, if we expand what's going on in our numerator here, OK, then notice that. We're gonna have 3. 6 squared X, OK. 362 X plus 6 cubed X, OK. minus times squared X multiplied by 6 X. And again, all of this is being divided by 3 + X squared. No, from our trigonometric identities, OK, we also know, or we can also recall, OK, that the square of tanics is equal to the square of 6 X minus 1. OK. So, with that idea in mind, if we apply that to our tangent, or squared tangent term here, that means we could rewrite it. As so we have 36 squared X plus 6 cubed X minus. 6 squared x minus 1 multiplied by 6 X. And again, all of this is being divided by 3 plus 6 X. 3 + X2. Now if we expand our term here, OK. Then we're gonna have 36 squared X plus 6 cubed X minus 6 cubed X minus 6 X. OK. And that's in our numerator right by our denominator. OK. And now if we take our numerator. Notice we can expand our negative term. We can distribute the negative. That's minus set cubed X plus set X, OK. And we remove our brackets. Now, set cubed X minus set cubed X equals 0. So then that means we're left with, let me put it in red, 36 squared X plus X, OK? All divided by 3 plus set X. Squared Thus this is the derivative of our term, the simplified form of the derivative of y. Thanks a lot for watching everyone. I hope this video helped.

Find the derivative of the following functions.
y = cot x / (1 + csc x)

Key Concepts

Derivative

Quotient Rule

Trigonometric Functions and Their Derivatives

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