Chapter 2: Problem 22
Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ g(t)=\frac{1}{\sqrt{t^{2}-2}} $$
Chapter 2: Problem 22
Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ g(t)=\frac{1}{\sqrt{t^{2}-2}} $$
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Get started for freeDifferential Equations In Exercises \(125-128\) , verify that the function satisfies the differential equation. $$ \text{Function} \quad \text{Differential Equation} $$ $$ y=2 \sin x+3 \quad y^{\prime \prime}+y=3 $$
Determining Differentiability In Exercises \(85-88\) , find the derivatives from the left and from the right at \(x=1\) (if they exist). Is the function differentiable at \(x=1 ?\) $$ f(x)=\left\\{\begin{array}{ll}{(x-1)^{3},} & {x \leq 1} \\ {(x-1)^{2},} & {x>1}\end{array}\right. $$
Determining Differentiability In Exercises \(75-80\) , describe the \(x\) -values at which \(f\) is differentiable. $$ f(x)=\left\\{\begin{array}{ll}{x^{2}-4,} & {x \leq 0} \\ {4-x^{2},} & {x>0}\end{array}\right. $$
Exploring a Relationship In Exercises 79 and \(80,\) verify that \(f^{\prime}(x)=g^{\prime}(x),\) and explain the relationship between \(f\) and \(g .\) $$ f(x)=\frac{3 x}{x+2}, g(x)=\frac{5 x+4}{x+2} $$
A ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 24 feet from the wall? (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.
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