Chapter 3.9, Problem 87E

a.

To determine

Whether the statement “The derivative of ${\log }_{2}9$ is $\frac{1}{9\ln 2}$” is true or not by explaining it or by giving a counter example.

b.

To determine

Whether the statement “$\ln \left(x+1\right)+\ln \left(x-1\right)=\ln \left(x^2-1\right),\text{for all }x$” is true or not by explaining it or by giving a counter example.

c.

To determine

Whether the following statement “The exponential function $2^{x+1}$ can be written in base e as $e^{2\ln \left(x+2\right)}$” is true or not by explaining it or by giving a counter example.

d.

To determine

To explain: The reason whether the following statement “$\frac{d}{dx}\left({\sqrt{2}}^x\right)=x{\sqrt{2}}^{x-1}$” is true or not by explaining it or by giving a counter example.

e.

To determine

To explain: The reason whether the following statement “$\frac{d}{dx}\left(x^{\sqrt{2}}\right)=\sqrt{2}{x}^{\sqrt{2}-1}$” is true or not by explaining it or by giving a counter example.

f.

To determine

To explain: The reason whether the following statement “${\left(4x+1\right)}^{\ln x}=x^{\ln \left(4x+1\right)}$” is true or not by explaining it or by giving a counter example.