Find the first derivative & second derivative of given functions w.r.t. corresponding independent variable.
`r=12/x`

Options

A. `(12)/(x^(2))`
B. `12x`
C. 12
D. `(-12)/(x^(2))`

Answer

To find the first and second derivatives of the function \( r = \frac{12}{x} \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ r = \frac{12}{x} \]
We can rewrite this as:
\[ r = 12x^{-1} \]

### Step 2: Find the first derivative
To find the first derivative \( \frac{dr}{dx} \), we apply the power rule of differentiation. The power rule states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \).

For our function:
\[ \frac{dr}{dx} = \frac{d}{dx}(12x^{-1}) = 12 \cdot (-1)x^{-2} = -\frac{12}{x^2} \]

### Step 3: Find the second derivative
Now, we need to find the second derivative \( \frac{d^2r}{dx^2} \). This involves differentiating the first derivative \( \frac{dr}{dx} = -\frac{12}{x^2} \).

We can rewrite the first derivative as:
\[ \frac{dr}{dx} = -12x^{-2} \]
Now, applying the power rule again:
\[ \frac{d^2r}{dx^2} = \frac{d}{dx}(-12x^{-2}) = -12 \cdot (-2)x^{-3} = \frac{24}{x^3} \]

### Final Results
Thus, the first derivative and second derivative of the function \( r = \frac{12}{x} \) with respect to \( x \) are:
\[ \frac{dr}{dx} = -\frac{12}{x^2} \]
\[ \frac{d^2r}{dx^2} = \frac{24}{x^3} \]

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