Question

If $y^3=x$ then the value of $\frac{dy}{dx}$ at $x=1$ is

• A. -3
• B. 3
• C. $\frac{1}{3}$
• D. $\frac{1}{\sqrt{3}}$

Hint:

For implicit differentiation, always remember to multiply by $\frac{dy}{dx}$ whenever you differentiate a term containing $y$. After differentiating, you'll have an equation involving $x, y,$ and $\frac{dy}{dx}$. Solve for $\frac{dy}{dx}$ algebraically. If you need to evaluate the derivative at a specific point, you'll often need to find the coordinates of both $x$ and $y$ at that point.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
This question asks for the derivative of an implicitly defined function at a specific point. We can use implicit differentiation to find an expression for $\frac{dy}{dx}$ and then substitute the given value of $x$ (and the corresponding value of $y$) to find the numerical value of the derivative. (Note: The original question in the source document is poorly scanned. The equation $y^3=x$ is a plausible interpretation that leads to one of the given options.)
Step 2: Key Formula or Approach
We will use implicit differentiation. We differentiate both sides of the equation $y^3=x$ with respect to $x$, remembering to apply the chain rule to terms involving $y$. \[ \frac{d}{dx}(y^3) = \frac{d}{dx}(x) \] \[ 3y^2 \frac{dy}{dx} = 1 \] Then we solve for $\frac{dy}{dx}$.
Step 3: Detailed Explanation
1. Differentiate the equation: Given the equation $y^3 = x$. Differentiating both sides with respect to $x$: \[ \frac{d}{dx}(y^3) = \frac{d}{dx}(x) \] Using the chain rule on the left side: \[ 3y^2 \cdot \frac{dy}{dx} = 1 \] 2. Solve for $\frac{dy{dx}$:} \[ \frac{dy}{dx} = \frac{1}{3y^2} \] 3. Find the value of y at x=1: We need to evaluate the derivative at $x=1$. We must first find the corresponding $y$ value from the original equation. When $x=1$, we have $y^3 = 1$, which means $y=1$. 4. Evaluate the derivative: Substitute $y=1$ into the expression for $\frac{dy}{dx}$: \[ \frac{dy}{dx} \bigg|_{x=1, y=1} = \frac{1}{3(1)^2} = \frac{1}{3} \] Step 4: Final Answer
The value of $\frac{dy}{dx}$ at $x=1$ is $\frac{1}{3}$.