Find $\frac{d(y^2)}{d(x^2)}$: Understand This Token

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In summary, "Find $\frac{d(y^2)}{d(x^2)}$" means finding the derivative of the function $y^2$ with respect to $x^2$ to determine the rate of change of $y^2$ with respect to $x^2$. The derivative is important because it helps us understand the slope or gradient of a function at a specific point. The derivative can be found using the chain rule or the product rule. $\frac{d(y^2)}{d(x^2)}$ is a second-order derivative, while $\frac{dy}{dx}$ is a first-order derivative and they are functions of $x^2$ and $x$ respectively.
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MHD93
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Let y = x^2 + 3x

and the question is Find [itex]\frac{dy^2}{dx^2}[/itex]

then how do I understand this token? is it [itex]\frac{d(y^2)}{d(x^2)}[/itex]
or [itex]\frac{(dy)^2}{(dx)^2}[/itex] = [itex](\frac{dy}{dx})^2[/itex]
or what ?

BTW: it's not wrong written, namely, not [itex]\frac{d^2y}{dx^2}[/itex]
 
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  • #2
The notation is unusual. I would guess that you let u=x2 and v=y2, and compute dv/du. Then put x and y back into the result.
 
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1. What does "Find $\frac{d(y^2)}{d(x^2)}$" mean?

The notation "Find $\frac{d(y^2)}{d(x^2)}$" is asking you to find the derivative of the function $y^2$ with respect to $x^2$. This means finding the rate of change of $y^2$ with respect to $x^2$.

2. Why is the derivative of $y^2$ with respect to $x^2$ important?

The derivative of $y^2$ with respect to $x^2$ is important because it tells us how the output, or $y$ value, of a function changes as the input, or $x$ value, changes. It helps us understand the slope or gradient of a function at a specific point.

3. How do I find the derivative of $y^2$ with respect to $x^2$?

The derivative of $y^2$ with respect to $x^2$ can be found using the chain rule. First, we rewrite the function as $y^2 = (y)^2$, and then we can use the power rule to find the derivative, which is $2y\frac{dy}{dx}$. We then substitute $\frac{dy}{dx}$ with $\frac{dy}{dx^2}$ to get the final answer of $2y\frac{dy}{dx^2}$.

4. Can I find the derivative of $y^2$ with respect to $x^2$ without using the chain rule?

Yes, the derivative of $y^2$ with respect to $x^2$ can also be found by rewriting the function as $y^2 = (y)^2$ and then using the product rule, which is $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$. In this case, $u = y$ and $v = y$, so the derivative becomes $\frac{dy}{dx^2}y + y\frac{dy}{dx^2}$. Simplifying this gives us the same answer of $2y\frac{dy}{dx^2}$.

5. How is $\frac{d(y^2)}{d(x^2)}$ different from $\frac{dy}{dx}$?

The notation $\frac{d(y^2)}{d(x^2)}$ is indicating that we are taking the derivative of $y^2$ with respect to $x^2$, while $\frac{dy}{dx}$ is indicating that we are taking the derivative of $y$ with respect to $x$. In other words, $\frac{d(y^2)}{d(x^2)}$ is a second-order derivative, while $\frac{dy}{dx}$ is a first-order derivative. Additionally, $\frac{d(y^2)}{d(x^2)}$ is a function of $x^2$, while $\frac{dy}{dx}$ is a function of $x$.

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