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Find the derivative of the following functions.$$y=\frac{a \sin x+b \cos x}{a \sin x-b \cos x} ; a \text { and } b \text { are nonzero constants }$$

$\frac{-2ab}{(a sin x - b cox x )^2}$

Calculus 1 / AB

Chapter 3

Derivatives

Section 5

Derivatives of Trigonometric Functions

Differentiation

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

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In mathematics, a derivati…

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In mathematics, a differen…

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Find the derivative of the…

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calculate the derivative w…

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Find the derivative. Simpl…

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Okay, so we're asked to find the derivative of why? Well, whenever y equals a sign of X plus B co sign of eggs all over a sign of X minus B co sign of X. So what I'm gonna end up doing is I'm gonna separate these out into effigy where f is going to be a sign X plus B coastline X and my G is gonna be a sign X minus PICO sine x Because I can do f over G. If I find the derivative of this, I can use my derivative formula Where have the derivative of f times G minus the derivative of G times F all over g squared. So let's hear on the side go ahead and find my derivative of F So I'm gonna break these down because of the derivative rules, I can actually do a sine X and then do be co sign X separately. So a times sign X so that is gonna carry down the derivative of Sine X is co sign X and bring down maybe the derivative of co sign of X is negative. Sign of X. So my answer is gonna be a co sign of X minus B sign of X because I can pull that negative in front of the beat and then repeating this with my G derivative. Again, a derivative of Sine X is co sign eggs minus B. And then again, my derivative of coastline is negative. Sign. Thanks. So I'm gonna have a co sign of X and then because I have a double negative, it becomes a positive. And just to make things simple, I'm actually gonna break down this formula even more so, What I'm going to do first is I am going to do F prime times G, and then we will put it together at the very yet. So my derivative is going to be a co sign X minus b sign of X times my G function, which is a sine x minus B co sign of X and let's go ahead and won't supply this up. So a times my A is going to be a squared sine x co sign X and then a times might be is gonna be minus 80 co sign at Squared X. Then I'm gonna repeat. So might be a sign of x times My A sign of X It's gonna be minus a b sine squared x and then my my to bees I'm gonna multiply together to get plus b squared sine x co sign X Okay, so now Minnick over here and do my G prime times f So my G prime is a co sign X plus b sine of x times my f which is a sine x plus b co sign of X So I'm gonna do my two A's together Multiply them so it's gonna be a squared sine x co sign X Gonna do my a co sign Times might be co sign to get a B co signed squared X The multiply might be signing my a sign together So plus a b science squared X and then multiply my to piece together So plus b squared sine x co sign ICS Okay, so now when I met up doing something according to the formula almost subtract my f prime g and my G prime f So it's going to be a squared sine x co sign X minus a p coast sine squared x minus A p sine squared x plus b squared sine X Co sign X and then I'm going to subtract my a squared. And if you notice I lined them up with those that were similar terms And since I am subtracting them, what I'm going to do is I'm gonna go through ants, put a a negative sign in each one of in front of each one to make the math a little bit easier so thes will end up cancelling out So my a squared sine x coastline Next wills will cancel because we have a plus and a minus so positive and negative the two a b co sine squared are going to both stay the same because they're both negative. Same thing with our science corpse. However r sine X and co sign X with r b squared are going to cancel. So if I go back to my f over g formula, you remember the difference which is this here? We're gonna combine these so my my two babies are not gonna be negative to a b co sine squared X. And then my two sine squared is minus two ap sine squared X all over my G squared, which is going to be a sine x minus B go sign X squared. And yes, this could be our final answer. However, we can simplify this down just a little bit. So if we notice in our numerator we have a negative to a B that I can pull out from each of these, which then leaves a coast sine squared eggs plus a sine squared eggs. If you remember from the trick functions coast sine squared X plus one plus signs cortex is going to give us one. So then our final answer B negative to AP over a sine x minus b Co sign X Square.

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