To find the derivative of the function y = ln(tan⁻¹(3x⁴)) with respect to x, we need to apply the chain rule. The chain rule states that the derivative of a composed function is the derivative of the outside function, times the derivative of the inside function. In this case, our outside function is ln(u) and our inside function is tan⁻¹(3x⁴).
1. Start by taking the derivative of the outside function. The derivative of ln(u) with respect to u is 1/u. So this gives us 1/tan⁻¹(3x⁴).
2. We then multiply this by the derivative of the inside function. The derivative of tan⁻¹(v) with respect to v is 1/(1+v^2).
3. Applying this to our inside function, we take the derivative of 3x⁴ with respect to x, which is 12x³.
4. Substituting the derivative we found for u and v back into our chain rule gives us:
```12x³/((1+ (3x⁴)²) * tan⁻¹(3x⁴))```
5. We simplify the equation by factoring out the square and rewriting it as:
```12x³/((9x⁸ + 1) * tan⁻¹(3x⁴))```
Therefore, the derivative of the given function y = ln(tan⁻¹(3x⁴)) with respect to x is:
```dy/dx = 12x³/((9x⁸ + 1) * tan⁻¹(3x⁴))```