Derivatives of Trig Functions: 동영상 및 연습문제

Find an equation of the line tangent to the curve $y=\cos \left(x\right)$ at $x=\frac{\pi }{2}$.

Evaluate $f^{\prime }\left(x\right)$ at $x=\frac{\pi }{4}$ if $f\left(x\right)=\cos x$.

Find the derivative of the function $y=3\cos x+4\sin x$.

Calculate the derivative of $y=\frac{\sin x+2}{\cos x}-7$.

Find the derivative of the function $f\left(x\right)=\frac{2\cos x+3\sin x}{2\cos x-3\sin x}$.

Find the derivative of $y={\left(2\sin x\right)}^2$.

Find the derivative of $y=\frac{\cos x}{2-\sin x}$.

Find the derivative of $y=\sec x+\csc x$.

Apply the Quotient Rule to find the derivative of the function $f\left(x\right)=\cot x$.

Determine the derivative of the function $f\left(x\right)=\tan x$ using the Quotient Rule.

Find the derivative of $y=\frac{\tan x}{3+\sec x}$.

A wave travels along a string, and its position in millimeters is described by $h\left(t\right)=15(2\cos t+1)$, for $t\ge 0$, where $t$ represents time in seconds. Determine the velocity of the wave, $v\left(t\right)=h^{\prime }\left(t\right)$.

A wave travels along a string, and its position in millimeters is described by $h\left(t\right)=15(2\cos t+1)$, for $t\ge 0$, where $t$ represents time in seconds. Determine the times and positions at which the velocity is zero.

Determine the limit of the function $f\left(x\right)=\frac{\sin x}{x-\pi }$ as $x\to \pi$.

Determine the limit of the function $f\left(x\right)=\frac{\sin x-1}{x-\frac{\pi }{2}}$ as $x\to \frac{\pi }{2}$.

Determine the values of $x$ when the slope of the given function is equal to 2.

$h\left(x\right)=2x-\cos x$

Draw the graph of the following function on $[0,2\pi ]$ and the tangent to the function at the given point.

$y=\frac{\cos x}{1-\cos x};x=\frac{\pi }{2}$

Using the identity $\cos 3x=4{\cos }^{3}x-3\cos x$, find the derivative $\frac{d}{dx}\left(\cos 3x\right)$. Then, use the identity $\sin 3x=3\sin x-4{\sin }^{3}x$ to express the derivative in terms of $\sin 3x$.

Consider the equation below. Using differentiation, state if it is true or false.

$\frac{d}{dx}(2\cot x{\csc }^{2}x-8\cot x-6x)=-6{\cot }^{4}x$

Determine the derivative of $z$ with respect to $w$, given $z=2w^4\tan w$.

Find the derivative of the function $y={\left(\frac{2\tan x}{1-\sin x}\right)}^{\frac{1}{2}}$.

Find the derivative of the function $y={\sec }^4\left(2x\right)$.

Find the derivative of $y=\sec \left(\tan x\right)$.

Find the derivative of the function $y=\cos \left(\sqrt{{\sin }^{2}x+4}\right)$.

A wave travels along a string, and its position in millimeters is described by $h\left(t_{}\right)=15(2\cos t+1)$, for $t\ge 0$, where $t$ represents time in seconds. Determine the acceleration function $a\left(t\right)$ of the wave and graph this function. Consider only one period in graphing.

Determine the equation of the line tangent to the graph of $y=1-2\cos x$ at $x=\frac{\pi }{3}$.

Determine the equation of the line tangent to the graph of $y=\sec x$ at $x=\frac{\pi }{3}$.

Consider the function $f\left(x\right)=\cos x$ on the interval $[0,2\pi ]$. Using the difference quotient $D\left(x\right)=\frac{f(x+0.02)-f\left(x\right)}{0.02}$, what does the graph of $D\left(x\right)$ resemble?

Two cyclists A and B are racing on a circular track. Their angular positions during the race are given by the functions $\alpha \left(t\right)$ for A and $\beta \left(t\right)$ for B, where $t$ is time in seconds and $0\le t\le 10$. At $t=0$, both cyclists are at the starting line, and at $t=10$, they have both completed one full circle. Which cyclist has the greater average angular velocity?

Calculate the derivative of the function $h\left(x\right)=(\sin x+\cos x)(\sin x-\cos x)$.