Find dy/dx given $y = 4x^3 – 1 + 2x^{1/2}$ where x > 0.

Question

I am unsure where the inequality x>0 comes into play in this question. Is it just there to confuse me? I differentiated the equation to get $12x^2+x^{-0.5}$. Is this the final answer or do I have to do something with the given inequality?

Answer 1

yes it is $$y'(x)=12x^2+2\cdot \frac{1}{2}x^{-1/2}$$

Answer 2

The inequality is only there because $x^{\frac{1}{2}}$ only exists for $x \geq 0$, and once you differentiate that you get, as you correctly found, $\frac{1}{2}x^{\frac{-1}{2}}$, which only exists for $x > 0$. So you've done it correctly.