91Ó°ÊÓ

Find the derivatives of the following functions. $$f(x)=\tanh ^{2} x$$

Oil consumption Starting in \(2018(t=0),\) the rate at which oil is consumed by a small country increases at a rate of \(1.5 \% / \mathrm{yr}\), start. ing with an initial rate of 1.2 million barrels/yr. a. How much oil is consumed over the course of the year 2018 (between \(t=0\) and \(t=1\) )? b. Find the function that gives the amount of oil consumed between \(t=0\) and any future time \(t\) c. How many years after 2018 will the amount of oil consumed since 2018 reach 10 million barrels?

Evaluate the following derivatives. $$\frac{d}{d x}\left(x^{\left(x^{10}\right)}\right)$$

Find the derivatives of the following functions. $$f(x)=\sqrt{\operatorname{coth} 3 x}$$

Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. Crime rate The homicide rate decreases at a rate of \(3 \% / \mathrm{yr}\) in a city that had 800 homicides/yr in 2018 . At this rate, when will the homicide rate reach 600 homicides/yr?

Find the derivatives of the following functions. $$f(x)=\ln \operatorname{sech} 2 x$$

Evaluate the following derivatives. $$\frac{d}{d x}\left(1+\frac{4}{x}\right)^{x}$$

Find the derivatives of the following functions. $$f(x)=x \tanh x$$

Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and units of time. Drug metabolism A drug is eliminated from the body at a rate of \(15 \% /\) hr. After how many hours does the amount of drug reach \(10 \%\) of the initial dose?

Evaluate the following derivatives. $$\frac{d}{d x}\left(\cos \left(x^{2 \sin x}\right)\right)$$