Let $ y = f(x) $ be a twice-differentiable function such that $ f(1) = 3 $ and $\frac{dy}{dx} = 4\sqrt{y^2 + 7x^2}$. What is the value of $\frac{d^2y}{dx^2}$ at $ x $?

A. 10
B. 23
C. 55
D. 160

Question

14-10-2023
Mathematics

High School

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Let $ y = f(x) $ be a twice-differentiable function such that $ f(1) = 3 $ and $\frac{dy}{dx} = 4\sqrt{y^2 + 7x^2}$. What is the value of $\frac{d^2y}{dx^2}$ at $ x $?

A. 10
B. 23
C. 55
D. 160

Answer :

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prabhatrathi24se prabhatrathi24se · Answer 1 · 2025-04-26T17:53:52-04:00

Final answer:

To calculate d²y/dx² for the given function, we differentiate the first derivative using the product rule and chain rule, but without knowing specific x or y values, we cannot determine the exact second derivative value from the options provided.

Explanation:

Given that y = f(x) is a twice-differentiable function where dy/dx = 4√(y² + 7x²), to find the second derivative d²y/dx² at x, we employ the chain rule and implicit differentiation. Since the first derivative dy/dx is already provided, we differentiate it with respect to x:

Let's first note that dy/dx is a function of both y and x. We need to take the derivative of both parts of the equation that depend on 'x', considering 'y' as a function of 'x'.

For the y² term inside the square root, we use the chain rule which states that the derivative of a function with respect to x is the derivative of the function with respect to y times the derivative of y with respect to x:

√(y² + 7x²) wrt x = (1/2)(y² + 7x²)^(-1/2)*(2yy'(dx/dx) + 14xx'(dx/dx))

Therefore, 4√(y² + 7x²) wrt x = 4*(1/2√(y² + 7x²)^(-1/2))*((2y*dy/dx) + 14x)

This simplifies to:

2((y)/(y² + 7x²)^(1/2))*(2y*dy/dx + 14x),

Now plug in the value of dy/dx = 4√(y² + 7x²) into the equation:

= 2*((y)/(y² + 7x²)^(1/2))*(2y*(4√(y² + 7x²)) + 14x)

We see that the terms with the square roots of (y² + 7x²) will cancel out, leaving:

= 4y*(2y) + 14x*2, which is 8y² + 28x.

However, the information provided does not give a specific value for y or x in order to calculate d²y/dx², thus we cannot definitively choose one of the given options A, B, C, or D.

Let \( y = f(x) \) be a twice-differentiable function such that \( f(1) = 3 \) and \(\frac{dy}{dx} = 4\sqrt{y^2 + 7x^2}\). What is the value of \(\frac{d^2y}{dx^2}\) at \( x \)?A. 10 B. 23 C. 55 D. 160

Answer :

Final answer:

Explanation:

Other Questions

Let \( y = f(x) \) be a twice-differentiable function such that \( f(1) = 3 \) and \(\frac{dy}{dx} = 4\sqrt{y^2 + 7x^2}\). What is the value of \(\frac{d^2y}{dx^2}\) at \( x \)?

A. 10
B. 23
C. 55
D. 160