Derivatives Calculation for $x^3 + e^x + 3^x + \cot x$

The question asks us to find the derivative of the function $f(x) = x^3 + e^x + 3^x + \cot x$ with respect to $x$. To solve this, we need to differentiate each term of the function separately using basic differentiation rules.

Standard Differentiation Rules

Here are the rules we'll use:

• The power rule: The derivative of $x^n$ is $nx^{n-1}$.
• The derivative of the exponential function $e^x$ is $e^x$.
• The derivative of an exponential function $a^x$ (where $a$ is a constant) is $a^x \ln(a)$.
• The derivative of $\cot x$ is $-\csc^2 x$.

Step-by-Step Differentiation

Let's differentiate each term of the function $f(x) = x^3 + e^x + 3^x + \cot x$:

1. Derivative of $x^3$: Using the power rule ($n=3$), the derivative is:

   $$ \frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2 $$

2. Derivative of $e^x$: The derivative of $e^x$ is simply $e^x$:

   $$ \frac{d}{dx}(e^x) = e^x $$

3. Derivative of $3^x$: Using the rule for $a^x$ where $a=3$, the derivative is:

   $$ \frac{d}{dx}(3^x) = 3^x \ln(3) $$

   (Note: $\ln(3)$ is often written as 'log 3' in contexts like the options provided).
4. Derivative of $\cot x$: The standard derivative of $\cot x$ is $-\csc^2 x$:

   $$ \frac{d}{dx}(\cot x) = -\csc^2 x $$

   (Note: $\csc^2 x$ is sometimes written as $\cos ec^2 x$).

Combining the Derivatives

Now, we add the derivatives of each term together:

$$ \frac{d}{dx}(x^3 + e^x + 3^x + \cot x) = \frac{d}{dx}(x^3) + \frac{d}{dx}(e^x) + \frac{d}{dx}(3^x) + \frac{d}{dx}(\cot x) $$

$$ = 3x^2 + e^x + 3^x \ln(3) - \csc^2 x $$

Final Derivative Expression

The final derivative of the given function is $3x^2 + e^x + 3^x \ln(3) - \csc^2 x$. This matches the expression found in Option 1.