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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Think About It Let \(f\) and \(g\) be functions whose first and second derivatives exist on an interval I. Which of the following formulas is (are) true? (a) \(f g^{\prime \prime}-f^{\prime \prime} g=\left(f g^{\prime}-f^{\prime} g\right)^{\prime} \quad\) (b) \(f g^{\prime \prime}+f^{\prime \prime} g=(f g)^{\prime \prime}\)
Differential Equations In Exercises \(125-128\) , verify that the function satisfies the differential equation. $$ \text{Function} \quad \text{Differential Equation} $$ $$ y=\frac{1}{x}, x>0 \quad x^{3} y^{\prime \prime}+2 x^{2} y^{\prime}=0 $$
Using Related Rates In Exercises \(1-4,\) assume that \(x\) and \(y\) are both differentiable functions of \(t\) and find the required values of \(d y / d t\) and \(d x / d t .\) $$ \begin{array}{rlrl}{y=\sqrt{x}} & {\text { (a) } \frac{d y}{d t} \text { when } x=4} & {} & {\frac{d x}{d t}=3} \\ {} & {\text { (b) } \frac{d x}{d t} \text { when } x=25} & {} & {\frac{d y}{d t}=2}\end{array} $$
Moving Point In Exercises \(5-8,\) a point is moving along the graph of the given function at the rate \(d x / d t .\) Find \(d y / d t\) for the given values of \(x .\) $$ \begin{array}{l}{y=\cos x ; \frac{d x}{d t}=4 \text { centimeters per second }} \\ {\begin{array}{llll}{\text { (a) } x=\frac{\pi}{6}} & {\text { (b) } x=\frac{\pi}{4}} & {\text { (c) } x=\frac{\pi}{3}}\end{array}}\end{array} $$
Determining Differentiability In Exercises \(75-80\) , describe the \(x\) -values at which \(f\) is differentiable. $$ f(x)=\sqrt{x-1} $$
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