The functions \(\fe{f}{x}=e^{3x}\text{,}\) \(\fe{g}{x}=\left(e^x\right)^3\text{,}\) and \(\fe{h}{x}=\left(e^3\right)^x\) are equivalent. Find the first derivative formulas for each of the functions (without altering their given forms) and then explicitly establish that the derivative formulas are the same.
Consider the function \(\fe{k}{\theta}=\fe{\sin^{-1}}{\fe{\sin}{\theta}}\text{.}\)
Over the interval \(\cinterval{-\frac{\pi}{2}}{\frac{\pi}{2}}\text{,}\) \(\fe{k}{\theta}=\theta\text{.}\) What does this tell you about the formula for \(\fe{\fd{k}}{\theta}\) over \(\ointerval{-\frac{\pi}{2}}{\frac{\pi}{2}}\text{?}\)
Use the chain rule and an appropriate trigonometric identity to verify your answer to Exercise 6.5.1.2. Please note that \(\fe{\cos}{\theta}\gt0\) over \(\ointerval{-\frac{\pi}{2}}{\frac{\pi}{2}}\text{.}\)
What is the constant value of \(\fe{\fd{k}}{\theta}\) over \(\ointerval{\frac{\pi}{2}}{\frac{3\pi}{2}}\text{?}\)
HintWhat is the value of \(\fe{\fd{k}}{\frac{\pi}{2}}\text{?}\)
Consider the function \(f\) shown in Figure 6.5.1.
Suppose that \(\fe{g}{x}=\left[\fe{f}{x}\right]^4\text{.}\) Over what intervals is \(\fd{g}\) positive?
Suppose that \(\fe{r}{x}=e^{\fe{f}{x}}\text{.}\) Over what intervals is \(\fd{r}\) positive?
Suppose that \(\fe{w}{x}=e^{\fe{f}{-x}}\text{.}\) Over what intervals is \(\fd{w}\) positive?
Suppose that \(\fe{h}{x}=\frac{1}{\fe{f}{x}}\text{.}\) Is \(h\) differentiable at \(-3\text{?}\) Please note that \(f\) has a horizontal tangent line at \(-3\text{.}\)
Decide whether or not it is necessary to use the chain rule when finding the derivative with respect to \(x\) of each of the following functions.
\(\fe{f}{x}=\fe{\ln}{x+1}\)
\(\fe{f}{x}=\dfrac{2}{x^5}\)
\(\fe{f}{x}=\fe{\cos}{\pi}\)
\(\fe{f}{x}=\fe{\cos}{x}\)
\(\fe{f}{x}=\fe{\cos}{\pi x}\)
\(\fe{f}{x}=\dfrac{\fe{\cos}{\pi x}}{\pi}\)
Find the first derivative with respect to \(x\) of each of the following functions. In all cases, look for appropriate simplifications before taking the derivative. Please note that some of the functions will be simpler to differentiate if you first use the rules of logarithms to expand and simplify the logarithmic expression.
\(\fe{f}{x}=\fe{\tan^{-1}}{\sqrt{x}}\)
\(\fe{f}{x}=e^{e^{\fe{\sin}{x}}}\)
\(\fe{f}{x}=\fe{\sin^{-1}}{\fe{\cos}{x}}\)
\(\fe{f}{x}=\fe{\tan}{x\fe{\sec}{x}}\)
\(\fe{f}{x}=\fe{\tan}{x}\fe{\sec}{\fe{\sec}{x}}\)
\(\fe{f}{x}=\sqrt[3]{\left(\fe{\sin}{x^2}\right)^2}\)
\(\fe{f}{x}=4x\fe{\sin^2}{x}\)
\(\fe{f}{x}=\fe{\ln}{x\fe{\ln}{x}}\)
\(\fe{f}{x}=\fe{\ln}{\dfrac{5}{xe^x}}\)
\(\fe{f}{x}=2\fe{\ln}{\sqrt[3]{x\fe{\tan^2}{x}}}\)
\(\fe{f}{x}=\fe{\ln}{\dfrac{e^{x+2}}{\sqrt{x+2}}}\)
\(\fe{f}{x}=\fe{\ln}{x^e+e}\)
\(\fe{f}{x}=\fe{\sec^4}{e^x}\)
\(\fe{f}{x}=\fe{\sec^{-1}}{e^x}\)
\(\fe{f}{x}=\fe{\csc}{\dfrac{1}{\sqrt{x}}}\)
\(\fe{f}{x}=\dfrac{1}{\fe{\csc}{\sqrt{x}}}\)
\(\fe{f}{x}=\dfrac{\fe{\tan^{-1}}{2x}}{2}\)
\(\fe{f}{x}=x^3\fe{\sin}{\dfrac{x}{3}}\)
\(\fe{f}{x}=\dfrac{4}{\sqrt{\frac{3}{x^7}}}\)
\(\fe{f}{x}=\dfrac{e^{xe^x}}{x}\)
\(\fe{f}{x}=xe^{xe^2}\)
\(\fe{f}{x}=\dfrac{\fe{\sin^5}{x}-\sqrt{\fe{\sin}{x}}}{\fe{\sin}{x}}\)
\(\fe{f}{x}=4x\fe{\sin}{x}\fe{\cos}{x^2}\)
\(\fe{f}{x}=\fe{\sin}{x\fe{\cos^2}{x}}\)
