True or False $\dfrac{d}{dx}\cos x= \sin x$

True or False $\dfrac{d}{dx}\tan x=$$\cot x$

True or False $\dfrac{d^{2}}{dx^{2}}\sin x=-\sin x$

True or False $\dfrac{d}{dx}\sin \dfrac{\pi }{3}=\cos \dfrac{\pi}{3}$

$f(x)=x-\sin x$, $c=\pi$

$f(\theta )=2\sin \theta +\cos \theta$, $c={\dfrac{\pi }{2}}$

$f(\theta )={\dfrac{\cos \theta }{1+\sin \theta }}$, $c=\dfrac{\pi }{3}$

$f(x)={\dfrac{\sin x}{1+\cos x}}$, $c=\dfrac{5\pi }{6}$

$y=3\sin \theta -2\cos \theta$

$y=4\tan \theta +\sin \theta$

$y=\sin x\cos x$

$y=\cot x\tan x$

$y=t\cos t$

$y=t^{2}\tan t$

$y=e^{x}\tan x$

$y=e^{x}\sec x$

$y=\pi \sec u\tan u$

$y=\pi u\tan u$

$y={\dfrac{\cot x}{x}}$

$y={\dfrac{\csc x}{x}}$

$y=x^{2}\sin x$

$y=t^{2}\tan t$

$y=t\tan t-\sqrt{3}\sec t$

$y=x\sec x+\sqrt{2}\cot x$

$y=\dfrac{\sin \theta }{1-\cos \theta}$

$y={\dfrac{x}{\cos x}}$

$y={\dfrac{\sin t}{1+t}}$

$y={\dfrac{\tan u}{1+u}}$

$y=\dfrac{\sin x}{e^{x}}$

$y=\dfrac{e^{x}}{\cos x}$

$y={\dfrac{\sin \theta +\cos \theta }{\sin \theta -\cos \theta }}$

$y={\dfrac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }}$

$y={\dfrac{\sec t}{1+t\sin t}}$

$y={\dfrac{\csc t}{1+t\cos t}}$

$y=\csc \theta\cot \theta$

$y=\tan \theta \cos \theta$

$y={\dfrac{1+\tan x}{1-\tan x}}$

$y={\dfrac{\csc x-\cot x}{\csc x+\cot x}}$

$y=\sin x$

$y=\cos x$

$y=\tan \theta$

$y=\sec \theta$

$y=t\sin t$

$y=t\cos t$

$y=e^{x}\sin x$

$y=e^{x}\cos x$

$y=2\sin u-3\cos u$

$y=3\sin u+4\cos u$

$y=a\sin x+b\cos x$

$y=a\sec\theta +b\tan \theta$

$f(x)=\sin x$ at $(0,0)$

$f(x)=\cos x$ at ${\left( \dfrac{\pi }{3},\dfrac{1}{2}\right) }$

$f(x)=\tan x$ at $(0,0)$

$f(x)=\tan x$ at ${\left( \dfrac{\pi }{4},1\right)}$

$f(x)=\sin x+\cos x$ at ${\left( \dfrac{\pi }{4},\sqrt{2}\right)}$

$f(x)=\sin x-\cos x$ at ${\left( \dfrac{\pi }{4},0\right) }$

$f(x)=2\sin x+\cos x$

$f(x)=\cos x-\sin x$

$f(x)=\sec x$

$f(x)=\csc x$

$f(x)=\sin x$

$f(\theta )=\cos \theta$

What is ${\lim\limits_{h\rightarrow 0}\dfrac{\cos \left( { \dfrac{\pi }{2}}+h\right) -\cos {\dfrac{\pi }{2}}}{h}?}$

What is $\lim\limits_{h\rightarrow 0}{\dfrac{\sin (\pi +h)-\sin \pi }{h}?}$

Simple Harmonic Motion The distance $s$ (in meters) of an object from the origin at time $t$ (in seconds) is modeled by the function $s( t) =\dfrac{1}{8}\cos t$.

Rate of Change A large, 8-ft high decorative mirror is placed on a wood floor and leaned against a wall. The weight of the mirror and the slickness of the floor cause the mirror to slip.

Sea Waves Waves in deep water tend to have the symmetric form of the function $f(x)=\sin x$. As they approach shore, however, the sea floor creates drag which changes the shape of the wave. The trough of the wave widens and the height of the wave increases, so the top of the wave is no longer symmetric with the trough. This type of wave can be represented by a function such as $$w(x)=\dfrac{4}{2+\cos x}.$$

Source: http://www.crd.bc.ca/watersheds/protection/geology-processes/waves.htm

Restaurant Sales A restaurant in Naples, Florida is very busy during the winter months and extremely slow over the summer. But every year the restaurant grows its sales. Suppose over the next two years, the revenue $R$, in units of $10,000, is projected to follow the model $$R=R( t) =\sin t+0.3t+1 \qquad 0 ≤ t ≤ 12$$ where $t=0$ corresponds to November 1, 2014; $t=1$ corresponds to January 1, 2015; $t=2$ corresponds to March 1, 2015; and so on.

If $y=\sin x$ and $y^{(n)}$ is the $n$th derivative of $y$ with respect to $x$ find the smallest positive integer $n$ for which $y^{(n)}=y$.

Use the identity $\sin A-\sin B=2\cos \dfrac{A+B}{2}\sin \dfrac{A-B}{2},$ with $A=x+h$ and $B=x$, to prove that $$\dfrac{d}{dx}\sin x= \lim\limits_{h\rightarrow 0}\dfrac{\sin (x+h)-\sin x}{h}=\cos x$$

Use the definition of a derivative to prove $\dfrac{d }{dx}\cos x=-\sin x$.

Derivative of ${y=\sec x}$ Use a derivative rule to show that $$\dfrac{d}{dx}\sec x=\sec x\tan x.$$

Derivative of $y=csc x$ Use a derivative rule to show that $$\dfrac{d}{dx}\csc x=-\csc x\cot x.$$

Derivative of ${y=\cot x}$ Use a derivative rule to show that $$\dfrac{d}{dx}\cot x=-\csc ^{2}x.$$

Let $f(x)=\cos x$. Show that finding $f^\prime (0)$ is the same as finding $\lim\limits_{x\rightarrow 0}{\dfrac{\cos x-1}{x}}$.

Let $f(x)=\sin x$. Show that finding $f^\prime (0)$ is the same as finding $\lim\limits_{x\rightarrow 0}{\dfrac{\sin x}{x}}$.

If $y=A\sin t+B\cos t$, where $A$ and $B$ are constants, show that $y''+y=0$.

For a differentiable function $f$, let $f^{\ast }$ be the function defined by $$f^{\ast }(x)=\lim\limits_{h\rightarrow 0}\frac{f(x+h)-f(x-h)}{h}$$