The total cost of daily output of x tons of coal is Rs. `((1)/(10) x^(3)-3x^(2) +50x)`. What is the Marginal cost funvtion .

Answer

To find the Marginal Cost function from the given total cost function, we will follow these steps:

### Step 1: Write down the total cost function
The total cost function (TC) is given as:
\[ TC(x) = \frac{1}{10} x^3 - 3x^2 + 50x \]

### Step 2: Differentiate the total cost function
To find the Marginal Cost (MC) function, we need to differentiate the total cost function with respect to \(x\):
\[ MC(x) = \frac{d(TC)}{dx} \]

### Step 3: Apply the differentiation rules
Now we will differentiate each term of the total cost function:
1. The derivative of \(\frac{1}{10} x^3\) is:
\[ \frac{3}{10} x^2 \]
2. The derivative of \(-3x^2\) is:
\[ -6x \]
3. The derivative of \(50x\) is:
\[ 50 \]

### Step 4: Combine the derivatives
Now, we combine the derivatives we calculated:
\[ MC(x) = \frac{3}{10} x^2 - 6x + 50 \]

### Step 5: Write the final Marginal Cost function
Thus, the Marginal Cost function is:
\[ MC(x) = \frac{3}{10} x^2 - 6x + 50 \]

### Summary
The Marginal Cost function derived from the total cost function is:
\[ MC(x) = \frac{3}{10} x^2 - 6x + 50 \]

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