Values of x that create parallel tangents in 2 curves

Question

Can anyone help me solve this?

I have been working on derivatives and I can’t solve this question

For what value(s) of $x$ do the curves $ y= (1 + x^3)^2$ and $y=2x^6$ have tangent so that are parallel?

Thanks

Answer 1

$$y=(1+x^3)^2, y=2x^6$$ Now,$$\frac{dy}{dx}=6x^2(1+x^3),\frac{dy}{dx}=12x^5$$

Since they are parallel, equate them: $$6x^2(1+x^3)=12x^5$$ $$1+x^3=2x^3$$ $$x^3(\frac{1}{x^3}+1)=2x^3$$ $$\frac{1}{x^3}=1$$ $$x^3=1$$ $$x=1$$