1
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1answer
29 views

Calculus - Trig Maximum Value Problem

When the rules of hockey were developed, Canada did not use the metric system. Thus, the distance between the goal posts was designated to be six feet. If Sidney Crosby is on the goal line, three feet ...
1
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0answers
25 views

Weierstrass function

I got stuck on this exercise from Prof. Tao's real analysis notes. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function $$f:= \sum_{n=1}^\infty 4^{-n} \sin(8^n\pi x)$$ Show that for every 8-dyadic ...
2
votes
2answers
54 views

Why are differential of $\sin^2(x)$ and integral of $\sin(2x)$ not the same?

I was working on a list of common integrals and differentials and I came across this question. If $${d\over d\theta}(\sin^2\theta) = \sin(2\theta)$$ Then why is $$\int \sin(2\theta) \space d\theta = ...
1
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0answers
43 views

Formula for nth derivative of $\arcsin^k(x/2)$

I need to find formula for $n$-th derivative of $\arcsin^k(\frac{x}{2})$. I have found formula for ...
0
votes
2answers
36 views

differentiation of tan(-x)

I've just started high school calculus. To differentiate trig functions the rule is $(f \circ g)' = g'(x) \cdot f'(g(x))$ So for $\tan(-x)$ would this not be $-\sec^2(-x)$? The answer says ...
-1
votes
2answers
26 views

What is the equation for a tangent to the graph of $y=\arcsin(x/2)$ at the origin?

I believe arc sin is the same as inverse sin but then I don't know how to deal with taking the derivative of that.
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1answer
28 views

Hard time with Derivatives of Inverse Functions

I'm having a really hard time with this question I keep googling for advice but can't find anything solid that's similar! Please help. I'm not sure if I should derive first or find the inverse first? ...
0
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0answers
10 views

Relation of Product Rule to sine sum identity

I was taking a look at the sine sum identity and noticed a resemblance to the Product Rule for derivatives. Applying this led to the following simplification: $$\begin{align}\sin(x + y) & = ...
0
votes
1answer
31 views

$-2(\sin x+2\cos 2x)=0$

I am finding the 2nd derivative critical values for graphing a trig function. So far I have it simplified to $$-2(\sin x+2\cos 2x)=0$$ What values for x make this equal zero? And is there a ...
0
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1answer
28 views

Critical Numbers Problems

Okay so I found the critical number no problem, it being cos x=-1/2, but on my answer sheet it says that the critical numbers are ...
15
votes
8answers
2k views

Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
0
votes
3answers
98 views

Derivatives of sine and cosine at $x=0$ give all values of $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$?

In video 3 of the video lectures by MIT on Single Variable Calculus presented by David Jerison, the latter says: Remarks: $\dfrac{d}{dx}\cos x\left|\right._{x=0}=\lim\limits_{\Delta ...
0
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3answers
72 views

Derivative of inverse function $\sin^{-1}(x)^2$

So $y=\sin^{-1}(x)^2$ I am asked to find $\frac{dy}{dx}$ Using the chain rule I find $\frac{dy}{dx}$= $2\sin^{-1}(x) * \frac{d}{dx}(\sin^{-1}(x))$ I let $z = \sin^{-1}(x)$ Multiplying both ...
1
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2answers
97 views

How to prove that $\frac{d}{dx}\sin(x)=\cos(x)$

I have to prove that $\dfrac{d}{dx}\sin(x)=\cos(x)$. I used the definition of a derivative: $$\dfrac{d}{dx}f(x)=\lim\limits_{h\to 0} \dfrac{f(x+h)-f(x)}{h}$$ $$\dfrac{d}{dx}\sin(x)=\lim\limits_{h\to ...
0
votes
1answer
21 views

I have a question regarding the relationship between tan(x) and sec(x).

This is a question that has been on my mind for sometime, and I'm getting two separate and contradictory answers to it. If $\tan x = 1$, then what will be the value of $\sec^2 x$? Now, one relation ...
0
votes
2answers
36 views

how to get $2/(t^2 + 1)$ as the derivative for Sin(theta) when $\tan(\theta/2) = t$

If $\sin \theta = \frac{2t}{1 + t^2}$ How do you get $d\theta = \frac{2}{1 + t^2}$ If you differentiate by quotient rule you get $\frac{2(1 - t^2)}{(1+t^2)^2}$ It is part of the solution to ...
0
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1answer
28 views

implicit differentiation using trigonometry functions

xcos(4x+3y)=ysinx I have been stuck on this problem for the longest. I have the answer but I don't know how to get to it. I have used the product and chain rule ...
0
votes
1answer
49 views

Minimize a trig function. Getting stuck.

So I have just about given up on this. Here is the problem. FYI, all angles are in degrees, and $L$, $R$ are just strictly positive scalars. I have a trig-function $D$. Its derivative shown below, ...
0
votes
1answer
24 views

Differentiability of trigonometric piecewise functions

So I have a function of a real variable $x$: $f(x) = \left\{\begin{array}{lr} x \int_0^{tanx} \dfrac{t^2}{\sqrt{1+t^3}}dt & if \: x \ge 0\\ sin^2(x) & if \: x \lt 0 ...
1
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2answers
23 views

$f(x) = \arccos {\frac{1-x^2}{1+x^2}}$; f'(0+), f'(0-)?

$f(x) = \arccos {\frac{1-x^2}{1+x^2}}$ $f'(x) = 2/(1+x^2)$, but I see graphic, and it is true only for x>=0. For x<=0 => $f'(x) = -2/(1+x^2)$ How can I deduce the second formula or proof that it ...
1
vote
1answer
49 views

Starting velocity by distance, time, and friction

I am writing a game in Javascript, and I just got a big math problem, where $\text{friction} = 0.97$. This is what is being looped every $1000$ / $60$ milliseconds, to make the projectile move all ...
2
votes
2answers
29 views

Finding conditions for a with given condition for critical points

$f(x)=\sin2x-8(a+1)\sin x+(4a^2+8a-14$)$x$. $x$ increases for all $x \in \mathbb{R}$ and has no critical points. Find values of $a$. My try: $f'(x)=4(\cos^2x-2(a+1)\cos x+a^2+2a-4)=0$ and ...
0
votes
2answers
121 views

Proof of Reduction Formula for $\displaystyle\int cos^n (x) \ dx = \frac{1}{n}\cos^{n-1}x\sin x + \frac{n-1}{n}\int\cos^{n-2}x \ dx$ [duplicate]

I ran into a question with proving the reduction formula: $\displaystyle\int cos^n x \ dx = \frac{1}{n}\cos^{n-1}x\sin x + \frac{n-1}{n}\int\cos^{n-2}x \ dx$ I then attempted to prove by ...
0
votes
2answers
131 views

Derivative of integral of $\sin (t^2)$

I'm stuck with the problem If $ F(x)=\int_0^{x^3} \sin t^2 dt$ find $F'(x)$ Now, if the upper interval were $x$, the answer would be $\sin t^2$ (right?). However, the upper interval is $x^3$. ...
1
vote
2answers
36 views

What is the derivate of $A \cdot\cos{ (B\cdot (x + C))} - D$?

$$f(x) = A \cdot \cos{(B\cdot(x+C))} - D$$ $$f'(x) = \text{ ?}$$ I would assume that the derivate is something like this: $$f'(x) = - A \cdot B \cdot \sin{( \dots )}$$ The thing that troubles me is ...
1
vote
2answers
97 views

Infinite derivatives of a trigonometric function

I've recently noticed that if you took an infinite amount of derivatives of a function, by that I mean something like this, $$ \lim_{x\to \infty} f^{'''\dots}(x)$$ Then if $f(x)$ is any polynomial ...
3
votes
3answers
106 views

How do you calculate the derivative after a change of variables?

How would you calculate $df \over dθ$ if $f(x,y) = x^2+y^2$ where $x = \sin 2θ$ and $y = \cos 2θ$? I tried Wolfram and using the product rule but I can't seem to get anywhere.
1
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2answers
93 views

Proving that $f(x) = \cos(x)\implies f'(x) = -\sin(x)$ using the definition of a derivative

I'm having trouble grasping the concept which proves that the derivative of $f(x) = \cos(x)$ is $f'(x) = -\sin(x)$. It needs to be proven using the definition of a derivative--and I can't quite piece ...
0
votes
2answers
42 views

Help on derivative of trigonometric functions

I have simple problem in derivative of trigonometric functions, can anyone show me the way to calculate the derivative of following function? $$y = \sin^2 x (\cos x + \sin x)$$ I think the answer ...
0
votes
1answer
54 views

Evaluate limit with L^Hopital's rule if possible

I'm not sure how to use the L'Hopital's rule properly. The question is $$ lim_{x\to \infty} \frac{x^2 + \sin x }{x^2} $$ I tried replacing x by $\frac{1}{t}$ and limit to $ lim_{t\to 0} $ and use ...
1
vote
1answer
52 views

Simplfiying a trigonometric polynomial

I was doing a derivative problem for calculus. The problem reads: $y=(\sec{x}+\tan{x})^5$ find $y'$. I have found a derivative, I believe is almost certainly correct as I have checked it with a ...
-1
votes
1answer
163 views

Cosine law derivative assistance needed

I read the wiki, but I would really appreciate it if someone can explain it to me and help me solve it; I don't know how to do it. By no means is this HW. Derivative Cosine law Given a planar ...
1
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1answer
48 views

Derivative of Trig. Function

if $f(x)=\tan(3x)$, then $f'(\pi/9)=$? I thought the answer was $4$ but my teacher marked it wrong. Work: $f'(x) = \sec^2(3x)\cdot 3 = \frac 3{\cos^2(3x)} = \frac{3}{\cos^2(\pi/3)} = 3/(3/4) = 4$.
1
vote
2answers
29 views

Find $\displaystyle \frac{dy}{dx}$ when, $\displaystyle y \arcsin x - x \arctan y = 1$

yesteryday was my class test and I found this question. Find $\displaystyle \frac{dy}{dx}$ when, $\displaystyle y \arcsin x - x \arctan y = 1$ I have read the question for arctanx as $1/1 + x^2$. ...
3
votes
4answers
110 views

What is $\frac{d(\arctan(x))}{dx}$?

Let $v= \arctan{x}$. Now I want to find $\frac{dv}{dx}$. My method is this: Rearranging yields $\tan(v) = x$ and so $dx = \sec^2(v)dv$. How do I simplify from here? Of course I could do something like ...
-1
votes
2answers
148 views

How to prove the equation $\cos x=2x$ has only one solution? [closed]

Show that the equation $\cos x=2x$ has only one solution, $x\in\mathbb{R}$.
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votes
1answer
31 views

Differentiate the functions trigonometry and derivatives

Differentiate $ln(\sec x+ \tan x)$ and $ (\sin x)^3\cos 3x+ (\cos x)^3\sin 3x$ with respect to $x$, simplifying where possible. Find the first and second derivatives (with respect to x) of the ...
1
vote
1answer
50 views

Derivative of trig function and limits

Find $$ \lim_{x\to \frac{\pi}{2}} \frac{tan(3x)}{tan(7x)} $$ I want to find it using l'hopital's rule My answer was : $$ \frac{\frac{sin(3x)}{cos(3x)}}{\frac{sin(7x)}{cos(7x)}} $$ $$ ...
0
votes
4answers
83 views

Derivative of trig function (l'hopital's rule)

Find: $$ \lim_{x\to 0} \frac{\sin^26x}{\sin^27x} $$ My answer was : $$ \frac{2\sin6x \cdot \cos6x}{2\sin7x \cdot \cos7x} $$ When I took the derivative of the numerator and denominator I got this $$ ...
1
vote
0answers
45 views

Derivative of trig function

Find the second derivative of $ \arcsin(2x^3) $ The solution says for the first derivative : $ \dfrac{1}{\sqrt{1-(2x^3)^2}} \cdot 6x^2 = \dfrac{6x^2}{\sqrt{1-4x^6}} $ When i answered the first ...
1
vote
2answers
40 views

Derivative of trigonometric function

How i can find the derivative of this trigonometric function $csc^4(8x^4-5)$ i tried to do it my self and i got to this $ 4[csc(8x^4-5)]^3 * [-csc(8x^4-5)cotan(8x^4-5)] $ The answer in the book ...
0
votes
3answers
770 views

Derivative of the sin(x) when x is measured in degrees

So a classic thing to derive in calculus textbooks is something like a statement as follows Is $\frac{d}{dx}\sin(u)$ the same as the derivative of $\frac{d}{dx}\sin(x)$ where $u$ is an angle measured ...
2
votes
0answers
75 views

Relating $x$ to $\frac{dx}{dt}$ in a right triangle.

I have a right triangle with sides of $x$ and $y$. I know $y$ is a constant (500) and that $\frac{d \theta}{dt}$ (where $\theta$ is the angle opposite from side $x$) is also constant ($8\pi$ rad/s). I ...
2
votes
5answers
97 views

Find $\lim_{x\to 0} \frac{\tan16x}{\sin2x}$

Find $\lim_{x\to 0} \frac{\tan16x}{\sin2x}$ I'm a little confused on limit trig. Am i suppose to simplify tan or do I use the derivative quotient rule? Please Help!!!
0
votes
1answer
50 views

Trigonometric functions properties

So I have a test next week and I have this question which I do know how to solve and I have no direction how to prove that. Any help would be appreciated. $$ \forall (x_{1}\land x_{2}) \in ...
1
vote
3answers
84 views

Differentiating w. r. t. $x$

Differentiate $$ \text{arccot} \frac{1-x}{1+x} $$ with respect to $x$ After putting $x= \cos \theta$, I got $$\text{arccot} \left(\tan^2 \frac{\theta}{2}\right)$$ Then how do I reach the answer? ...
0
votes
1answer
116 views

Trigonometric Functions. Definite Integrals

Find, correct to one decimal place, the value of $$\int_{0}^{60} 2\sin(x/2) \, dx.$$ Can someone please show me how this question is done. It would be very helpful thanks!
2
votes
2answers
80 views

Simplifying second derivative using trigonometric identities

Given that $x=1+\sin(t)$ , $y=\sin(t) -\frac{1}{2} \cos(2t)$ show that $\frac{\text{d}^2y}{\text{d}x^2}=2$. I am having trouble proving this. Here is my working so far: ...
1
vote
1answer
131 views

Higher Derivatives of trigonometric functions

The position of a particle is given by $s = 5 \cos (2t+ (\pi/4))$ at time $t$ . What are the maximum values of the displacement,the velocity,and the acceleration? The answers are displacement: $5$ ...
5
votes
1answer
377 views

Finding the derivative of $\sin \sqrt {x^2+1}$ from the definition?

This means finding $\lim_{h \to 0} \large \large \frac{\sin \sqrt {(x+h)^2+1}-\sin \sqrt {x^2+1}}{h}$ . The only way I could think of to do this is to replace $h$ by some function $f(h)$ such that ...