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Chapter 2: Problem 56
Use the alternative form of the derivative to find the derivative at \(x=c\) (if it exists). \(g(x)=(x+3)^{1 / 3}, \quad c=-3\)
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In Exercises \(75-80\), evaluate the derivative of the function at the indicated point. Use a graphing utility to verify your result. \(\frac{\text { Function }}{y=\frac{1}{x}+\sqrt{\cos x}} \quad \frac{\text { Point }}{\left(\frac{\pi}{2}, \frac{2}{\pi}\right)}\)
In Exercises 107-110, (a) use a graphing utility to find the derivative of the function at the given point, (b) find an equation of the tangent line to the graph of the function at the given point, and (c) use the utility to graph the function and its tangent line in the same viewing window. \(s(t)=\frac{(4-2 t) \sqrt{1+t}}{3},\left(0, \frac{4}{3}\right)\)
A television camera at ground level is filming the lift-off of a space shuttle at a point 750 meters from the launch pad. Let \(\theta\) be the angle of elevation of the shuttle and let \(s\) be the distance between the camera and the shuttle (as shown in the figure). Write \(\theta\) as a function of \(s\) for the period of time when the shuttle is moving vertically. Differentiate the result to find \(d \theta / d t\) in terms of \(s\) and \(d s / d t\).
Let \(f\) be a differentiable function of period \(p\). (a) Is the function \(f^{\prime}\) periodic? Verify your answer. (b) Consider the function \(g(x)=f(2 x)\). Is the function \(g^{\prime}(x)\) periodic? Verify your answer.
Prove that \(\arcsin x=\arctan \left(\frac{x}{\sqrt{1-x^{2}}}\right),|x|<1\)
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