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Chapter 5: Problem 96
Define the base for the natural logarithmic function.
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Finding the Maximum Rate of Change Verify that the function $$ y=\frac{L}{1+a e^{-x / b}}, \quad a>0, \quad b>0, \quad L>0 $$ increases at a maximum rate when \(y=L / 2\)
Find the derivative of the function. \(y=\arctan x+\frac{x}{1+x^{2}}\)
Chemical Reactions Chemicals \(A\) and \(B\) combine in a 3 -to- 1 ratio to form a compound. The amount of compound \(x\) being produced at any time \(t\) is proportional to the unchanged amounts of \(A\) and \(B\) remaining in the solution. So, when 3 kilograms of \(A\) is mixed with 2 kilograms of \(B\) , you have $$\frac{d x}{d t}=k\left(3-\frac{3 x}{4}\right)\left(2-\frac{x}{4}\right)=\frac{3 k}{16}\left(x^{2}-12 x+32\right)$$ One kilogram of the compound is formed after 10 minutes. Find the amount formed after 20 minutes by solving the equation $$\int \frac{3 k}{16} d t=\int \frac{d x}{x^{2}-12 x+32}$$
Find the derivative of the function. \(g(x)=3 \arccos \frac{x}{2}\)
Prove 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) \(\frac{d}{d x}[\operatorname{arccsc} u]=\frac{-u^{\prime}}{|u| \sqrt{u^{2}-1}}\)
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