91Ó°ÊÓ

Evaluate the following derivatives. $$\frac{d}{d x}\left((\ln 2 x)^{-5}\right)$$

Evaluate the following derivatives. $$\frac{d}{d x}\left(\ln ^{3}\left(3 x^{2}+2\right)\right)$$

Identities Prove each identity using the definitions of the hyperbolic functions. \(\tanh (-x)=-\tanh x\)

Identities Prove each identity using the definitions of the hyperbolic functions. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x(\text {Hint}:\) Begin with the right side of the equation.)

Absolute and relative growth rates Two functions \(f\) and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant. $$f(t)=100+10.5 t, \quad g(t)=100 e^{t / 10}$$

Evaluate the following derivatives. $$\frac{d}{d x}\left((2 x)^{4 x}\right)$$

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

Absolute and relative growth rates Two functions \(f\) and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant. $$f(t)=2200+400 t, \quad g(t)=400 \cdot 2^{t / 20}$$

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

Designing exponential growth functions Complete the following steps for the given situation. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. Answer the accompanying question. Population The population of a town with a 2016 population of 90,000 grows at a rate of \(2.4 \% /\) yr. In what year will the population reach \(120.000 ?\)