In the following find the approximate values, using differentials :
`sqrt(0.6).`

Answer

To find the approximate value of \(\sqrt{0.6}\) using differentials, we can follow these steps:

### Step 1: Identify a nearby value
Choose a value of \(x\) that is close to \(0.6\) and for which we can easily compute the square root. Here, we can choose \(x = 0.64\) because \(\sqrt{0.64} = 0.8\).

### Step 2: Define the function
Let \(y = f(x) = \sqrt{x}\).

### Step 3: Calculate \(dy\)
We need to find \(dy\) using the derivative of \(f(x)\). The derivative of \(f(x) = \sqrt{x}\) is given by:
\[ f'(x) = \frac{1}{2\sqrt{x}} \]

### Step 4: Evaluate the derivative at \(x = 0.64\)
Now, we evaluate the derivative at \(x = 0.64\):
\[ f'(0.64) = \frac{1}{2\sqrt{0.64}} = \frac{1}{2 \cdot 0.8} = \frac{1}{1.6} \]

### Step 5: Calculate \(dx\)
Next, we find \(dx\), which is the change in \(x\) from \(0.64\) to \(0.6\):
\[ dx = 0.6 - 0.64 = -0.04 \]

### Step 6: Calculate \(dy\)
Now we can find \(dy\):
\[ dy = f'(0.64) \cdot dx = \frac{1}{1.6} \cdot (-0.04) \]
Calculating this gives:
\[ dy = -\frac{0.04}{1.6} = -0.025 \]

### Step 7: Approximate the value of \(\sqrt{0.6}\)
Using the differential approximation:
\[ \sqrt{0.6} \approx y + dy = 0.8 - 0.025 = 0.775 \]

### Final Answer
Thus, the approximate value of \(\sqrt{0.6}\) is \(0.775\).

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