prove-that-fracdd-xsec-xsec-x-tan-x

Question

Prove that $\frac{d}{d x}(\sec x)=\sec x 	an x$

Nicholas Barvinok · Accepted Answer

we want to prove that the derivative with respect to X d over DX of the function, seek and X is equal to seek inns X times Tangent X Well, let&#x27;s just try the most direct approach by just taking the derivative of Seek and X well, the derivative of seeking Ex. It might be easier to first right out in terms of signs and coastlines. Seeking X is one over co sign X, and when you see that you might be tempted to use the quotient rule. But that&#x27;s one of the more complicated derivative rules. And if you see that instead, you can write this as co sign next to the minus one. Then it becomes easier to differentiate. So we have co sign X to the minus one. This is a composition of functions. It has an inter function co sign X, and it has the outer function of raising co sign next to the minus one. So by the change by the chain rule, the first thing we do is we take the derivative of the outer function, raising costa next to the minus one, which by the power rule, tells us that we bring the minus one down and, uh um subtract one from the exponents of coastline X. So from minus one, it becomes a minus two. And now again by the chain rule, we multiply by the derivative of the Inter Function CO sign X, which is minus sine X, and we can simplify this a bit. We have two minus signs So those cancel out and we end up getting sign X over co sign Squared X and we want to get seeking access times Tangent X Well, we can get a sequined by bringing out one of the co signs in the denominator one of her co sign X And then what do we have left? You are left with precisely sign over co sign and thus we get seeking ex Tangent X. We can write a little box two in form the reader that we have reached the end of the proof

Amrita Bhasin · Answer

this question asked us to prove that the derivative of Kasey Kahne axe is negative. Caustic in Taxco tension axe. Obviously, this is a trick. No metric identity, but we&#x27;re asked to prove it. So we&#x27;re gonna be using the question rule off one G minus F U one over g squared. Okay, so let&#x27;s take the derivative. The derivative is gonna be a sign of acts Time zero, because that&#x27;s the derivative of a constant minus one times co sign of axe over sine squared of X. This simplifies to be negative coastline of acts over sine squared effects which simplifies to be negative co tangent. Excuse. Remember, code tenderness coast on over sign times cause seeking X, which is one oversight because remember, we have an extra sign on the bottom. Therefore, we have now proven that both sides are equivalent

Grant Mansfield · Answer

All right. So the question of are given today is we need to prove that De of d X. So the derivative of coasts he can&#x27;t x is equal to negative. Cosi can&#x27;t x times co tangent axe. And that&#x27;s what we&#x27;re going to try and do. All we have to do is prove that that sign equals right side. So the first step that we&#x27;re gonna do is we&#x27;re going to put, um, coast. He can&#x27;t ex into a bit more relative terms. So maybe D of d. X and we want to use the reciprocal identity soco seek an X is actually equal to, um won over Sign X. And now what we can dio after this step is we can use product roll. So by using one is our f and sign as RG, and we&#x27;re just gonna plug it into the formula and kind of see what happens. So, using the formula, we still want this. Um, no. By using this for a story, she by using this formula, we&#x27;re going to get something that looks like this. So we have our sine x times zero, which is the derivative of, um one minus one times the derivative of sign acts, which is Khost X and all that is going to be over sine X squared. So now that this is done, we can actually simplify this. We simplify this too negative. Khost X over sine squared x. And now that we use this, we can also use the reciprocal rule to, um swift this up and, um, make this negative one over sine x times coasts over or Khost X over sine X. So all we did we have. So the sine squared x um right here actually really means, um sine x times sign X. So what we did was we broke those up. We made one over sine X, and then coast acts over sine X. We multiplied those together we get the same answer is the step above Now using this, we can actually, um, see that as we stated before, um, using different colors here one over sine X that is equal to Khost, seek an X and another trick identity that should be known in order to solve this question is that Khost X over sine X is equal to co tangent X meaning. If we simplify these problems, we&#x27;re going to get this final answer. Cosi can&#x27;t x times, Coty Engine X If we look back up at the start, we can see that&#x27;s the answer that we wanted. So therefore, we proved that the derivative of coast he can&#x27;t X is equal to co seeking X CO two changing axe.

Shamshad Waris · Answer

and maybe Bob close eggs is it Will do. Minus Alex. So I do very, very deep formula for the any puncture. If our necks did you quit, let me Del extends to dio if our X plus bill eggs minus it. Well, eggs divided by Daleks applied this from black hair. Then they get it. Very, very Bob or six the place off. Aquatics here. The cause. It&#x27;s so we can I let me bill extent todo they oppose explicit Bill X minus for six. Divided well daily? No, we can do one thing, and they will be, Well, both eggs remit Bill extends to Geo. Yeah, I don&#x27;t know the cause. A plus B. Well, because into cause me minus signing to send. So apply here. Then he&#x27;ll get course eggs into cause bill X minus Sign exit. Do saying bill eggs minus four sits divided by buildings. Now we can simplify. Take this time Course x include Alex. Nine is boss eggs then right sine X in tow. Sang the legs. Do I didn&#x27;t like buildings? No. Take X comin from the past down So the cost excrement from the first down. Then you can be pain Waas X minus one, minus sighing eggs into sane Bilic&#x27;s divide every day lives No, I think that done here. Then they get limit. They extend to you course it into cause bill X minus one Divided way Bill eggs minus sign eggs in do sighing Bill Ex Vice Bilic&#x27;s as you know, limit off course Exposed Bell X minus one by Delic Is Do so we&#x27;ll get here X into Julia minus sine x the limiter Science. They&#x27;ll exploit Dilek City. But getting the formula then here yet u minus saying so. Then delivered a tip off course X equal minus signings fruit and you.

Zachary Mitchell · Answer

Let&#x27;s go ahead and start with the left hand side. That&#x27;s 10 of X times Cosi can&#x27;t of X And now let&#x27;s go ahead and rewrite this equation. Using the definitions of Tana Backs and Co ck Innovex we have. This is equal to sign of X divided by co sign of X times one over sign of X. Now notice the science Cancel out. We&#x27;re left with one over CO sign X, and this, by definition, is equal to seek out of X. That&#x27;s it.

Amrita Bhasin · Answer

this question asks us to prove that the derivative of co tangent acts is negative. Cassie can&#x27;t squared X. We&#x27;re gonna be using the quotient rule for this, which is F one G minus F G one over G squared. In other words, we&#x27;re gonna be differentiating co Tension X, which is equivalent to coastline of acts over sign of X. So we&#x27;re gonna be differentiating co sign over sign. Okay, First off sign stays the same. The derivative of coastline is negative. Sign of acts were then subtracting co sign of X. The derivative of sign is co sign of Axe divided by G squared. This is sine squared of axe. This simplifies to be negative one over sine squared of acts which is the same thing as negative Kaseke unscored of acts. The reason wise because one over sign is Cassie can&#x27;t therefore negative one over Science Court of X is negative. Costigan scored of X. Therefore, we have now proven both side

Problem

Prove that $\frac{d}{d x}(\cot x)=-\csc ^{2} x$

Problem 46 Medium Difficulty

Prove that $\frac{d}{d x}(\sec x)=\sec x \tan x$

Answer

First write sec $x$ as $\frac{1}{\cos x}$ then differentiate using quotient rule.

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James Stewart

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First write sec $x$ as $\frac{1}{\cos x}$ then differentiate using quotient rule.

Biocalculus Calculus for the Life Sciences

Grace H.

Heather Z.

Kristen K.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Grace H.

Heather Z.

Kristen K.

Joseph L.

James Stewart

Biocalculus Calculus for the Life Sciences