If variables x and y are related by the equation
`x=int_(0)^(y)(1)/(sqrt(1+9u^(2)))` du, 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

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

we will use the Leibniz rule for differentiation under the integral sign.

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

According to the Leibniz rule, if we have an integral of the form

\[ x = \int_{a}^{b(y)} f(u) \, du, \]

then the derivative with respect to \(y\) is given by:

\[ \frac{dx}{dy} = f(b(y)) \cdot \frac{db}{dy} - f(a) \cdot \frac{da}{dy}. \]

In our case, \(a = 0\), \(b(y) = y\), and \(f(u) = \frac{1}{\sqrt{1 + 9u^2}}\).

### Step 2: Apply the Leibniz rule

Substituting into the Leibniz rule, we have:

\[ \frac{dx}{dy} = f(y) \cdot \frac{dy}{dy} - f(0) \cdot \frac{d0}{dy}. \]

Calculating \(f(y)\) and \(f(0)\):

1. \(f(y) = \frac{1}{\sqrt{1 + 9y^2}}\)
2. \(f(0) = \frac{1}{\sqrt{1 + 9 \cdot 0^2}} = \frac{1}{\sqrt{1}} = 1\)

Since \(\frac{d0}{dy} = 0\), we can simplify:

\[ \frac{dx}{dy} = \frac{1}{\sqrt{1 + 9y^2}} \cdot 1 - 1 \cdot 0 = \frac{1}{\sqrt{1 + 9y^2}}. \]

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

Now, we need to find \(\frac{dy}{dx}\). We know that:

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

Taking the reciprocal gives us:

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

### Final Answer

Thus, the value of \(\frac{dy}{dx}\) is:

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

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