2
votes
1answer
54 views

What is an intuitive way to see $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?

Without calculation, explain why $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?
1
vote
1answer
14 views

Inverse Laplace transform, none factorable denominator

I am really stumpted on this problem and can't seem to figure out where to go from where I am. Can anyone give me some advice or hint where I should do next? Here is the problem: ...
0
votes
4answers
69 views

Derivative with respect of a function

i have a function of two variables: $f(\theta,\phi) = \theta \sin(\phi)$ and i have to differentiate $f(\theta,\phi)$ with respect to: $1 - 0.5\theta^2$ That is: ...
0
votes
2answers
42 views

Problem related to Mean Value Theorem

I found out a question that I can't figure out a way to solve it. Plz can anyone help me. Question is, Prove that $\exists\,C\in(0,\pi/4)\,\mathrm{s.t.}\,\tan(\pi/4+C)=3/C$ I know this should be ...
2
votes
3answers
65 views

Using the Chain Rule to prove trig derivatives

I'm having trouble with this problem, I'm not sure how to tackle it and I was wondering if somebody could set me on the right path. The problem is as follows: Use the Chain Rule to show that if ...
1
vote
2answers
58 views

Find the derivative of $y=\cos(x) - 2\sin(x),$ when the gradient is $1$

I need to find the smallest positive value of $x$ for which the gradient of the curve has value 1. For this equation: $$ y =\cos(x)-2\sin(x) $$ The answer is 2.5c grad. The following is my ...
0
votes
1answer
17 views

Discovering the derivatives of functions combined with trig values.

Hey StackExchange I have a problem that I don't really understand and I could use some hints for starting it. Suppose $m(\frac{\pi}{3}) = 4$ and $ m'(\frac{\pi}{3}) = -2$, and let $g(x) = m(x)\sin x$ ...
1
vote
1answer
35 views

Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
5
votes
1answer
103 views

Where did I go wrong on trying to solve this question on an exam?

I took an exam yesterday, and I almost for a fact know I got this question wrong. I couldn't figure it out, since my answer wasn't an answer choice, so I ended up guessing. An explanation of what I ...
2
votes
2answers
44 views

How to resolve multiply differentiation function algorithms?

My simple function is $f(x)=\frac{1}{2}e^{-x}\sin(2x)$; Can I resolve for multiply differentiation $f^{(n)}=?$ algorithm? Thx for answer.
1
vote
2answers
49 views

How do I go about solving this derivative of inverse tangent?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$8\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=8\frac{1}{1+x^2}$$would ...
1
vote
1answer
31 views

How do I solve this trig derivative in respect to $x$?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=\frac{1}{1+x^2}$$would ...
3
votes
2answers
233 views

I am having problems figuring out how to derive this.

I have the function $$\tag{1} f(x)=\ln\sqrt{8+\cos^2x}$$ So we derive it as follows: $$\tag{2} f(x)=\ln(8+\cos^2x)^\frac{1}{2}$$ $$\tag{3} f(x)=\frac{1}{2}\ln(8+\cos^2x)$$ $$\tag{4} ...
1
vote
2answers
100 views

How do I go about solving this derivative?

I have the function $$f(x)=\ln\sqrt{8+\cos^2x}$$ so $$1.f(x)=\ln(8+\cos^2x)^\frac{1}{2}$$so$$2.f(x)=\frac{1}{2}\ln(8+\cos^2x)$$so $$3.f'(x)=\frac{1}{2}\left[\frac{-2 \cos x^{\sin x}}{8+\cos ...
0
votes
1answer
37 views

Derivative of a Trigonometric Function Help

Trying to derive a trigonometric function, Wolfram Alpha and my textbook provide two different answers. Here is the function: $$y = {\cot x\over (1+\csc x)}$$ First step using quotient rule results ...
1
vote
2answers
64 views

Help finding the second derivative of this function.

I need help finding the second derivative of this function. I found the first derivative and the second, but the program says my answer is incorrect either by typing error and it won't process ...
0
votes
2answers
37 views

If $y=2\sin^{-1}\sqrt{1-x}+\sin^{-1}(2\sqrt{x(1-x)})$ for $0<x<\displaystyle\frac{1}{2}$ then what is the value of $\displaystyle\frac{dy}{dx}$

If $y=2\sin^{-1}\sqrt{1-x}+\sin^{-1}(2\sqrt{x(1-x)})$ for $0<x<\displaystyle\frac{1}{2}$ then $\displaystyle\frac{dy}{dx}$ equals : A) $\displaystyle\frac{2}{\sqrt{x(1-x)}}$ B) ...
0
votes
1answer
38 views

I need help on deriving this trig function.

I am trying to derive $e^x \sin x - 2x \csc x$. I tried using the product and difference rule. So I got the derivative for $e^x \sin x$ and got $(e^x)(\cos(x))+(\sin(x))(e^x)$ and for the derivative ...
1
vote
4answers
85 views

Local minimum of $f(x) = 4x + \frac{9\pi^2}{x} + \sin x$

What's the minimum value of the function $$f(x) = 4x + \frac{9\pi^2}{x} + \sin x$$ for $0 < x < +\infty$? The answer should be $12\pi - 1$, but I get stuck with the expression involving both ...
1
vote
4answers
61 views

Need an example of piece wise function continuous but not differentiable

I Need an example of piece wise function continuous but not differentiable. One of the functions has to be trigonometric and the other has to be exponential. Please
2
votes
3answers
48 views

Limits of trig functions

How can I find the following problems using elementary trigonometry? $$\lim_{x\to 0}\frac{1−\cos x}{x^2}.$$ $$\lim_{x\to0}\frac{\tan x−\sin x}{x^3}. $$ Have attempted trig identities, didn't help. ...
2
votes
1answer
82 views

How to derive $\frac{d}{dx}\left(x+1\right)^{\sin\left(x\right)}$

I need help to find derivative of: $\frac{d}{dx}(x+1)^{\sin x}$ i tried to do something like this.. $$(x+1)^{\sin x}\cdot \ln\left(x+1\right)=\sin x(x+1)^{\sin\left(x\right)-1}\cdot ...
1
vote
3answers
99 views

Why is $\cos(x)$ the derivative of $\sin(x)$?

The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $\cos(x)$ is $-\sin(x)$. Is there a simple proof of this, preferably using pictures?
1
vote
3answers
44 views

$\frac d{dx}\cos x \space\mathrm{vs}\space\frac d{dx}\cos(-x)$

Like it says on the tin. I thought that the rule for deriving $sin{x}$ and $\cos x$ was simply the chain rule; $\displaystyle\frac d{dx}f(g(x))=f'(g(x))g'(x)$; applying to $\cos(x)$, this appears to ...
0
votes
2answers
37 views

derivative of trigonometry for cos to the power of 3

This appears in my homework and I don't know how to do it, could you help me please? $f(x) = \cos ^3 (4x + 1)$ with $0 < x < 1 $ Find the derivative of $f(x)$. I know the derivative of $\cos ...
1
vote
1answer
39 views

Prove that $\frac{\pi}{2}-x<\tan^{-1}(x)<\frac{\pi}{2}-x+\frac{x^3}{3}$

Prove that for every $x>0$, it is true: $$\frac{\pi}{2}-x<\tan^{-1}(x)<\frac{\pi}{2}-x+\frac{x^3}{3}$$ We can split it into two statements: $\frac{\pi}{2}-x<\tan^{-1}(x)$ ...
4
votes
5answers
175 views

$99$th derivative of $\sin x$

Can someone help me calculate the $99$th derivative of $\sin(x)$? Calculate $f^{(99)}(x) $ for the function $f(x) = \sin(x) $
0
votes
1answer
54 views

Differentiation/ find the derivative

Can anybody please help me with my work? I have to find the differentiate/ find the derivative of these two question: Please HELP!!! $sin^2(cos3x^3)^5 $ $cot^2(x)((x^2)(3cos^3(3x)))^2$
1
vote
1answer
138 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 ...
2
votes
1answer
55 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
58 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
vote
0answers
91 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
41 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
48 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.
0
votes
1answer
46 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
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
votes
1answer
30 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
3k 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
140 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
votes
3answers
114 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
vote
2answers
105 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
25 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
44 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
votes
1answer
38 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
62 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
50 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
vote
2answers
29 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
58 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
35 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
182 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 ...