Mathematical Exploration
The Problem
We are asked to find the derivative of the function:
The notation $\cot^{1}$ is often ambiguous. For this solution, we will interpret it as the inverse cotangent function (also written as $\operatorname{arccot}$ or $\cot^{-1}$). If a different interpretation was intended, the solution would change accordingly.
So, we interpret the function as:
The Solution: Finding $\frac{dy}{dx}$
To find the derivative $\frac{dy}{dx}$, we'll use the chain rule.
Step 1: Identify the outer and inner functions.
Let the inner function be $u = 2x^3$.
Then the outer function becomes $y = \operatorname{arccot}(u) + 1$.
Step 2: Differentiate the inner function.
The derivative of $u$ with respect to $x$ is:
Step 3: Differentiate the outer function.
The derivative of $y$ with respect to $u$ is:
Recall that $\frac{d}{du}(\operatorname{arccot}(u)) = -\frac{1}{1+u^2}$.
The derivative of the constant $1$ is $0$.
Step 4: Apply the Chain Rule.
The chain rule states: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.
Step 5: Substitute back the inner function.
Substitute $u = 2x^3$ back into the expression:
Simplify the expression:
Final Answer:
The derivative of $y = \operatorname{arccot}(2x^3) + 1$ is: