As always, you want to simplify an expression before jumping in to take its derivative. Nevertheless, it can build confidence to see that the rules work even if you don't simplify first.

# Subsection6.3.1Exercises

Consider the functions $\fe{f}{x}=\sqrt{x^2}$ and $\fe{g}{x}=\left(\sqrt{x}\right)^2\text{.}$

##### 1

Assuming that $x$ is not negative, how does each of these formulas simplify? Use the simplified formula to find the formulas for $\fe{\fd{f}}{x}$ and $\fe{\fd{g}}{x}\text{.}$

##### 2

Use the chain rule (without first simplifying) to find the formulas for $\fe{\fd{f}}{x}$ and $\fe{\fd{g}}{x}\text{;}$ simplify each result (assuming that $x$ is positive).

##### 3

Are $f$ and $g$ the same function? Explain why or why not.

##### 4

So long as $x$ falls on the interval $\ointerval{-\frac{\pi}{2}}{\frac{\pi}{2}}\text{,}$ $\fe{\tan^{-1}}{\fe{\tan}{x}}=x\text{.}$ Use the chain rule to find $\lzoo{x}{\fe{\tan^{-1}}{\fe{\tan}{x}}}$ and show that it simplifies as it should.

##### 5

How does the formula $\fe{g}{t}=\fe{\ln}{e^{5t}}$ simplify and what does this tell you about the formula for $\fe{\fd{g}}{t}\text{?}$ After answering those questions, use the chain rule to find the formula for $\fe{\fd{g}}{t}$ and show that it simplifies as it should.

Consider the function $\fe{g}{t}=\fe{\ln}{\frac{5}{t^3\fe{\sec}{t}}}\text{.}$

##### 6

Find the formula for $\fe{\fd{g}}{t}$ without first simplifying the formula for $\fe{g}{t}\text{.}$

##### 7

Use the quotient, product, and power rules of logarithms to expand the formula for $\fe{g}{t}$ into three logarithmic terms. Then find $\fe{\fd{g}}{t}$ by taking the derivative of the expanded version of $g\text{.}$

##### 8

Show the two resultant fomulas are in fact the same. Also, reflect upon which process of differentiation was less work and easier to “clean up.”