삼각함수의 미분

For the following exercises, find $\frac{{d}^{2}y}{d{x}^{2}}$ for the given functions.

$y=\frac{1}{x}+\tan x$

For the following exercises, find $\frac{{d}^{2}y}{d{x}^{2}}$ for the given functions.

$y=2\csc x$

For the following exercises, find $\frac{{d}^{2}y}{d{x}^{2}}$ for the given functions.

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

Find all $x$ values on the graph of $f(x)=-3\sin x\cos x$ where the tangent line is horizontal.

Find all $x$ values on the graph of $f(x)=x-2\cos x$ for $0<x<2\pi$ where the tangent line has slope 2.

[T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function $s(t)=-6\cos t$ where $s$ is measured in inches and $t$ is measured in seconds. Find the rate at which the spring is oscillating at $t=5$ s.

Let the position of a swinging pendulum in simple harmonic motion be given by $s(t)=a\cos t+b\sin t$ where $a$ and $b$ are constants, $t$ measures time in seconds, and $s$ measures position in centimeters. If the position is 0 cm and the velocity is 3 cm/s when $t=0$, find the values of $a$ and $b$.

After a diver jumps off a diving board, the edge of the board oscillates with position given by $s(t)=-5\cos t$ cm at $t$ seconds after the jump.

a. Sketch one period of the position function for $t\ge 0.$ b. Find the velocity function. c. Sketch one period of the velocity function for $t\ge 0.$ d. Determine the times when the velocity is 0 over one period. e. Find the acceleration function. f. Sketch one period of the acceleration function for $t\ge 0.$

The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by $y=10+5\sin x$ where $y$ is the number of hamburgers sold and $x$ represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find $y'$ and determine the intervals where the number of burgers being sold is increasing.

[T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by $y(t)=0.5+0.3\cos t,$ where $t$ is months since January. Find ${y}^{'}$ and use a calculator to determine the intervals where the amount of rain falling is decreasing.

For the following exercises, use the quotient rule to derive the given equations.

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

For the following exercises, use the quotient rule to derive the given equations.

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

For the following exercises, use the quotient rule to derive the given equations.

$\frac{d}{dx}(\csc x)=-\csc x\cot x$

Use the definition of derivative and the identity

$\cos (x+h)=\cos x\cos h-\sin x\sin h$ to prove that $\frac{d(\cos x)}{dx}=-\sin x.$

For the following exercises, find the requested higher-order derivative for the given functions.

$\frac{{d}^{3}y}{d{x}^{3}}$ of $y=3\cos x$

For the following exercises, find the requested higher-order derivative for the given functions.

$\frac{{d}^{2}y}{d{x}^{2}}$ of $y=3\sin x+{x}^{2}\cos x$

For the following exercises, find the requested higher-order derivative for the given functions.

$\frac{{d}^{4}y}{d{x}^{4}}$ of $y=5\cos x$

For the following exercises, find the requested higher-order derivative for the given functions.

$\frac{{d}^{2}y}{d{x}^{2}}$ of $y=\sec x+\cot x$

For the following exercises, find the requested higher-order derivative for the given functions.

$\frac{{d}^{3}y}{d{x}^{3}}$ of $y={x}^{10}-\sec x$

Graph sine and cosine on the same axes, from $x=-2\pi$ to $x=2\pi$. Mark the points where $\sin x$ has a horizontal tangent. What do these points correspond to, on the graph of cosine?