If `x= int_(0)^(y) (dt)/(sqrt(1+9t^(2)))` then `(dy)/(dx)` is equal to

Options

A. `(1)/(sqrt(1+9y^(2)))`
B. `sqrt(1+9y^(2))`
C. `(1+ 9y^(2))`
D. `(1)/(1+ 9y^(2))`

Answer

To find \(\frac{dy}{dx}\) given the equation

\[ x = \int_{0}^{y} \frac{dt}{\sqrt{1 + 9t^2}}, \]

we will use the Fundamental Theorem of Calculus and implicit differentiation. Here are the steps to solve the problem:

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

We start with the equation:

\[ x = \int_{0}^{y} \frac{dt}{\sqrt{1 + 9t^2}}. \]

Differentiating both sides with respect to \(x\):

\[ \frac{dx}{dx} = \frac{d}{dx} \left( \int_{0}^{y} \frac{dt}{\sqrt{1 + 9t^2}} \right). \]

### Step 2: Apply the Fundamental Theorem of Calculus

Using the Fundamental Theorem of Calculus, we differentiate the integral:

\[ 1 = \frac{d}{dy} \left( \int_{0}^{y} \frac{dt}{\sqrt{1 + 9t^2}} \right) \cdot \frac{dy}{dx}. \]

This gives us:

\[ 1 = \frac{1}{\sqrt{1 + 9y^2}} \cdot \frac{dy}{dx}. \]

### Step 3: Solve for \(\frac{dy}{dx}\)

Now, we can isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = \sqrt{1 + 9y^2}. \]

### Final Answer

Thus, we have:

\[ \frac{dy}{dx} = \sqrt{1 + 9y^2}. \]

---