Let `y=x^3-8x+7a n dx=f(t)dot` If `(dy)/(dt)=2` and `x=3` at `t=0,` then `(dx)/(dt)` at `t=0` is given by 1 (b) `(19)/2` (c) `2/(19)` (d) none of these

Options

A. 1
B. `(19)/(2)`
C. `(2)/(19)`
D. None of these

Explanation

To solve the problem step by step, we will use the chain rule of differentiation. Let's break it down:

### Step 1: Understand the relationship between the variables
We have:
- \( y = x^3 - 8x + 7 \)
- \( x = f(t) \)
- Given \( \frac{dy}{dt} = 2 \) and \( x = 3 \) when \( t = 0 \).

### Step 2: Differentiate \( y \) with respect to \( t \)
Using the chain rule, we can express \( \frac{dy}{dt} \) as:
\[ \frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt} \]

### Step 3: Find \( \frac{dy}{dx} \)
First, we need to differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^3 - 8x + 7) = 3x^2 - 8 \]

### Step 4: Substitute \( x = 3 \) into \( \frac{dy}{dx} \)
Now, we will substitute \( x = 3 \) into \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = 3(3^2) - 8 = 3(9) - 8 = 27 - 8 = 19 \]

### Step 5: Substitute into the chain rule equation
Now we know \( \frac{dy}{dt} = 2 \) and \( \frac{dy}{dx} = 19 \). We can substitute these values into the chain rule equation:
\[ 2 = 19 \cdot \frac{dx}{dt} \]

### Step 6: Solve for \( \frac{dx}{dt} \)
Now, we can solve for \( \frac{dx}{dt} \):
\[ \frac{dx}{dt} = \frac{2}{19} \]

### Conclusion
Thus, the value of \( \frac{dx}{dt} \) at \( t = 0 \) is:
\[ \frac{2}{19} \]

### Answer
The correct option is (c) \( \frac{2}{19} \).

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