Find derivative of given functions w.r.t the respective independent variable .
`q=sqrt(2r-r^(2))`, find `(dq)/(dr)`

Answer

To find the derivative of the function \( q = \sqrt{2r - r^2} \) with respect to \( r \), we will use the chain rule and the power rule of differentiation. Here’s a step-by-step solution:

### Step 1: Rewrite the function
We start with the function:
\[ q = \sqrt{2r - r^2} \]
This can be rewritten using exponent notation:
\[ q = (2r - r^2)^{1/2} \]

### Step 2: Apply the chain rule
To differentiate \( q \) with respect to \( r \), we apply the chain rule:
\[ \frac{dq}{dr} = \frac{1}{2}(2r - r^2)^{-1/2} \cdot \frac{d}{dr}(2r - r^2) \]

### Step 3: Differentiate the inner function
Now we need to differentiate the inner function \( 2r - r^2 \):
\[ \frac{d}{dr}(2r - r^2) = 2 - 2r \]

### Step 4: Substitute back into the derivative
Now we substitute this result back into our derivative expression:
\[ \frac{dq}{dr} = \frac{1}{2}(2r - r^2)^{-1/2} \cdot (2 - 2r) \]

### Step 5: Simplify the expression
Now we can simplify the expression:
\[ \frac{dq}{dr} = \frac{2 - 2r}{2\sqrt{2r - r^2}} \]
This can be further simplified to:
\[ \frac{dq}{dr} = \frac{1 - r}{\sqrt{2r - r^2}} \]

### Final Answer
Thus, the derivative of \( q \) with respect to \( r \) is:
\[ \frac{dq}{dr} = \frac{1 - r}{\sqrt{2r - r^2}} \]

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