What is the rate of change of demand (x) of a commodity with respect to it’s price (y) If `y = 10 + x + 25x^(3)`

Options

A. `10/(1+ 75x^(2))`
B. `1/(1+75x^(2))`
C. `1+ 75x^(2)`
D. `-1/(1+75x^(2))`

Answer

To find the rate of change of demand \( x \) of a commodity with respect to its price \( y \), given the equation:

\[ y = 10 + x + 25x^3 \]

we need to follow these steps:

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

We start by differentiating the equation with respect to \( x \):

\[ \frac{dy}{dx} = \frac{d}{dx}(10) + \frac{d}{dx}(x) + \frac{d}{dx}(25x^3) \]

Calculating each term:

- The derivative of \( 10 \) is \( 0 \).
- The derivative of \( x \) is \( 1 \).
- The derivative of \( 25x^3 \) is \( 75x^2 \) (using the power rule).

Putting it all together:

\[ \frac{dy}{dx} = 0 + 1 + 75x^2 = 1 + 75x^2 \]

### Step 2: Find \( \frac{dx}{dy} \)

Now, we need to find the rate of change of \( x \) with respect to \( y \), which is the reciprocal of \( \frac{dy}{dx} \):

\[ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{1 + 75x^2} \]

### Final Answer

Thus, the rate of change of demand \( x \) with respect to its price \( y \) is:

\[ \frac{dx}{dy} = \frac{1}{1 + 75x^2} \]