Chapter 2: Problem 21
Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=\frac{1}{\sqrt{3 x+5}} $$
Chapter 2: Problem 21
Finding a Derivative In Exercises \(7-34,\) find the derivative of the function. $$ y=\frac{1}{\sqrt{3 x+5}} $$
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Get started for freeLet \(k\) be a fixed positive integer. The \(n\) hh derivative of \(\frac{1}{x^{k}-1}\) has the form \(\frac{P_{n}(x)}{\left(x^{k}-1\right)^{n+1}}\) where \(P_{n}(x)\) is a polynomial. Find \(P_{n}(1) .\)
A spherical balloon is inflated with gas at the rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 centimeters (b) 60 centimeters?
Linear and Quadratic Approximations The linear and quadratic approximations of a function \(f\) at \(x=a\) are $$\begin{array}{l}{P_{1}(x)=f^{\prime}(a)(x-a)+f(a) \text { and }} \\\ {P_{2}(x)=\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}+f^{\prime}(a)(x-a)+f(a)}\end{array}$$ In Exercises 123 and \(124,\) (a) find the specified linear and quadratic approximations of \(f,(b)\) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\) . $$ f(x)=\sec x ; \quad a=\frac{\pi}{6} $$
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 \(x=1 ?\) $$ f(x)=\sqrt{1-x^{2}} $$
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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