If $y=x-x^3+3$ find the rate of change of y with respect to $x^2$ as a function of x

Question

If $y=x-x^3+3$ find the rate of change of y with respect to $x^2$ as a function of x?

$$y=x-x^3+3$$

$$\frac{d}{dx^2}\left(y=x-x^3+3\right)$$

Is it $-3x^2$, because of the power rule? Let say I subsititute $z$ for $x^2$

So, $\frac{d}{dx^2}\left(z-z^3+3\right)=1-3z^2$, then I substitute back $z=x^2$. So $1-3\left(x^2\right)^2$, which is $1-3\cdot x^4$?

Answer 1

Let $p = x^2$, then we can rewrite the equation as $y = p^{1/2} - p^{3/2} + 3$.

Taking the derivative with respect to $p$, we get $\frac{dy}{dp} = (1/2)p^{-1/2} - (3/2)p^{1/2}$.

Substituting $x^2$ for $p$, we get $\frac{dy}{dx^2} = (1/2){(x^2)}^{-1/2} - (3/2){(x^2)}^{1/2} = \frac{1}{2x} - \frac{3x}{2}$.

Answer 2

Let $u=x^2$ where $du=2 x dx$. Then $\frac{d}{dx^2}f(x)=\frac{d}{du}f(x)=\frac{d}{2 x dx} f(x)=\frac{1}{2x} \cdot \frac{d}{dx}(f(x))$