Chapter 2: Problem 48
Find the derivative of the function. \(y=x^{2} e^{-x}\)
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Evaluate the second derivative of the function at the given point. Use a computer algebra system to verify your result. \(f(x)=\frac{1}{\sqrt{x+4}}, \quad\left(0, \frac{1}{2}\right)\)
Verify each differentiation formula. (a) \(\frac{d}{d x}[\arctan u]=\frac{u^{\prime}}{1+u^{2}}\) (b) \(\frac{d}{d x}[\operatorname{arccot} u]=\frac{-u^{\prime}}{1+u^{2}}\) (c) \(\frac{d}{d x}[\operatorname{arcsec} u]=\frac{u^{\prime}}{|u| \sqrt{u^{2}-1}}\) (d) \(\frac{d}{d x}[\arccos u]=\frac{-u^{\prime}}{\sqrt{1-u^{2}}}\) (e) \(\frac{d}{d x}[\operatorname{arccsc} u]=\frac{-u^{\prime}}{|u| \sqrt{u^{2}-1}}\)
Prove that \(\arcsin x=\arctan \left(\frac{x}{\sqrt{1-x^{2}}}\right),|x|<1\)
In Exercises \(75-80\), evaluate the derivative of the function at the indicated point. Use a graphing utility to verify your result. \(\frac{\text { Function }}{y=37-\sec ^{3}(2 x)} \quad \frac{\text { Point }}{(0,36)}\)
Given that \(g(5)=-3, \quad g^{\prime}(5)=6, \quad h(5)=3,\) and \(h^{\prime}(5)=-2,\) find \(f^{\prime}(5)\) (if possible) for each of the following. If it is not possible, state what additional information is required. (a) \(f(x)=g(x) h(x)\) (b) \(f(x)=g(h(x))\) (c) \(f(x)=\frac{g(x)}{h(x)}\) (d) \(f(x)=[g(x)]^{3}\)
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