If `x=int_(0)^(y)(1)/(sqrt(1+4t^(2)))` dt, then `(d^(2)y)/(dx^(2))`, is

Options

A. 2y
B. 4y
C. 8y
D. 6y

Answer

To find \(\frac{d^2y}{dx^2}\) given that

\[ x = \int_0^y \frac{1}{\sqrt{1 + 4t^2}} \, dt, \]

we will follow these steps:

### Step 1: Differentiate \(x\) with respect to \(y\)

Using the Fundamental Theorem of Calculus, we differentiate both sides with respect to \(y\):

\[ \frac{dx}{dy} = \frac{1}{\sqrt{1 + 4y^2}}. \]

### Step 2: Differentiate \(y\) with respect to \(x\)

To find \(\frac{dy}{dx}\), we take the reciprocal of \(\frac{dx}{dy}\):

\[ \frac{dy}{dx} = \sqrt{1 + 4y^2}. \]

### Step 3: Differentiate \(\frac{dy}{dx}\) with respect to \(x\)

Now, we need to differentiate \(\frac{dy}{dx}\) with respect to \(x\) to find \(\frac{d^2y}{dx^2}\). We apply the chain rule:

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\sqrt{1 + 4y^2}\right). \]

Using the chain rule, we have:

\[ \frac{d^2y}{dx^2} = \frac{1}{2\sqrt{1 + 4y^2}} \cdot \frac{d}{dx}(1 + 4y^2). \]

### Step 4: Differentiate \(1 + 4y^2\) with respect to \(x\)

Using the chain rule again:

\[ \frac{d}{dx}(1 + 4y^2) = 8y \frac{dy}{dx}. \]

### Step 5: Substitute \(\frac{dy}{dx}\) into the equation

Now substitute \(\frac{dy}{dx} = \sqrt{1 + 4y^2}\):

\[ \frac{d^2y}{dx^2} = \frac{1}{2\sqrt{1 + 4y^2}} \cdot 8y \sqrt{1 + 4y^2}. \]

### Step 6: Simplify the expression

Simplifying gives us:

\[ \frac{d^2y}{dx^2} = \frac{8y}{2} = 4y. \]

### Final Result

Thus, the final result for \(\frac{d^2y}{dx^2}\) is:

\[ \frac{d^2y}{dx^2} = 4y. \]

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