Question

What is the derivative of cot^2x^3

Answer 1

To find the derivative of \( \cot^2(x^3) \), we will apply the chain rule and the derivative of the cotangent function.

Let \( y = \cot^2(x^3) \).

We can rewrite this as:
\[
y = (\cot(x^3))^2
\]

Now, we will use the chain rule:
\[
\frac{dy}{dx} = 2 \cot(x^3) \cdot \frac{d}{dx}[\cot(x^3)]
\]

Next, we need to find \( \frac{d}{dx}[\cot(x^3)] \). Using the chain rule again:

The derivative of \( \cot(u) \) is \( -\csc^2(u) \cdot \frac{du}{dx} \), where \( u = x^3 \). So, we have:
\[
\frac{d}{dx}[\cot(x^3)] = -\csc^2(x^3) \cdot \frac{d}{dx}(x^3) = -\csc^2(x^3) \cdot 3x^2
\]

Putting it all together, we find:
\[
\frac{dy}{dx} = 2 \cot(x^3) \cdot \left(-\csc^2(x^3) \cdot 3x^2\right)
\]
\[
= -6x^2 \cot(x^3) \csc^2(x^3)
\]

Therefore, the derivative of \( \cot^2(x^3) \) is:
\[
\frac{dy}{dx} = -6x^2 \cot(x^3) \csc^2(x^3)
\]