Given the function:

$f(x) = \ln(3x - \tan x) - e^{8x^3} + 2^x$

Let's break down the structure of the function $f(x)$:

1. Components of the Function:

• $\ln(3x - \tan x)$: This logarithmic part requires that the argument $3x - \tan x$ be positive, meaning $3x > \tan x$. Also, $x$ must be in a domain where the tangent function is defined (not at odd multiples of $\frac{\pi}{2}$).

• $e^{8x^3}$: An exponential function that grows very rapidly as $x$ increases, especially since the exponent is cubic in $x$.

• $2^x$: Another exponential function, which also increases rapidly, though not as fast as $e^{8x^3}$ for large values of $x$.

2. Domain of $f(x)$:

• For $\ln(3x - \tan x)$ to be defined, $3x - \tan x > 0$ and $x \neq \frac{(2n+1)\pi}{2}$ for any integer $n$ because $\tan x$ is undefined at those points.

3. Finding the Derivative:

To find the derivative $f'(x)$, we use the following rules:

• The derivative of $\ln(g(x))$ is $\frac{g'(x)}{g(x)}$.
• The derivative of $e^{g(x)}$ is $e^{g(x)} \cdot g'(x)$.
• The derivative of $2^x$ is $2^x \ln 2$.

Applying these rules:

[ f'(x) = \frac{d}{dx}[\ln(3x - \tan x)] - \frac{d}{dx}[e^{8x^3}] + \frac{d}{dx}[2^x] ]

Let's compute each part:

1. $\frac{d}{dx}[\ln(3x - \tan x)]$:

   • Let $g(x) = 3x - \tan x$.
   • $g'(x) = 3 - \sec^2 x$.
   • So, $\frac{d}{dx}[\ln(3x - \tan x)] = \frac{3 - \sec^2 x}{3x - \tan x}$.

2. $\frac{d}{dx}[e^{8x^3}]$:

   • Let $h(x) = 8x^3$.
   • $h'(x) = 24x^2$.
   • So, $\frac{d}{dx}[e^{8x^3}] = e^{8x^3} \cdot 24x^2$.

3. $\frac{d}{dx}[2^x]$:

   • The derivative of $2^x$ is $2^x \ln 2$.

Putting it all together:

$f'(x) = \frac{3 - \sec^2 x}{3x - \tan x} - 24x^2 e^{8x^3} + 2^x \ln 2$

Would you like a further breakdown of the derivative or an exploration of the properties of this function?

5 Relative Questions to Expand Understanding:

1. What is the domain of the function $f(x) = \ln(3x - \tan x) - e^{8x^3} + 2^x$?
2. How does the presence of exponential terms affect the growth rate of $f(x)$ as $x$ increases?
3. What are the potential points of discontinuity in $f(x)$ given the logarithmic and tangent functions?
4. How would you find the critical points of this function?
5. Can this function have any local maxima or minima given the rapid growth of the exponential components?

Tip:

When dealing with logarithmic functions, always check the domain to ensure the argument of the logarithm is positive and valid throughout the desired range of $x$.