Question:

Step 1: Identify the function and its properties. We are given: $$f(x) = x^4 \tan x^3 - x \ln(1 + x^2)$$ We need to find the fourth derivative of $f(x)$ evaluated at $x = 0$, that is, $\dfrac{d^4 f(x)}{dx^4}\bigg|_{x=0}$. Step 2: Determine whether $f(x)$ is an odd or even function. Let us evaluate $f(-x)$: $$f(-x) = (-x)^4 \tan((-x)^3) - (-x)\ln(1 + (-x)^2)$$ Simplifying each term: - First term: $(-x)^4 \tan(-x^3) = x^4 \cdot (-\tan x^3) = -x^4 \tan x^3$ (since $\tan$ is an odd function) - Second term: $-(-x)\ln(1 + x^2) = x\ln(1 + x^2)$ Therefore: $$f(-x) = -x^4 \tan x^3 + x\ln(1 + x^2) = -\left(x^4 \tan x^3 - x\ln(1 + x^2)\right) = -f(x)$$ This confirms that $f(x)$ is an **odd function**. Step 3: Apply the property of derivatives of odd functions. For an odd function $f(x)$ where $f(-x) = -f(x)$, we can establish that all even-order derivatives at $x = 0$ must equal zero. This is because if $f$ is odd, then $f'$ is even, $f''$ is odd, $f'''$ is even, and $f^{(4)}$ is odd. More specifically, the even-order derivatives (0th, 2nd, 4th, 6th, ...) of an odd function vanish at $x = 0$. Step 4: Evaluate the fourth derivative at $x = 0$. Since the fourth derivative is an even-order derivative and $f(x)$ is an odd function: $$\dfrac{d^4 f(x)}{dx^4}\bigg|_{x=0} = f^{(4)}(0) = 0$$ **Final Answer:** The value of $\dfrac{d^4 f(x)}{dx^4}$ at $x = 0$ is $\boxed{0}$. This corresponds to **Option 1**.