Given `u=7t^(4)-2t^(3)-8t-5, " find " (du)/(dt)`.
Hence, find `(du)/(dt)` at t = 0, 1, 2.

Answer

To solve the problem, we need to find the derivative of the function \( u \) with respect to \( t \) and then evaluate this derivative at \( t = 0, 1, \) and \( 2 \).

### Step 1: Differentiate \( u \)
Given:
\[ u = 7t^4 - 2t^3 - 8t - 5 \]

To find \( \frac{du}{dt} \), we differentiate each term of \( u \) with respect to \( t \):

1. The derivative of \( 7t^4 \) is \( 7 \cdot 4t^{4-1} = 28t^3 \).
2. The derivative of \( -2t^3 \) is \( -2 \cdot 3t^{3-1} = -6t^2 \).
3. The derivative of \( -8t \) is \( -8 \).
4. The derivative of \( -5 \) is \( 0 \).

Putting it all together, we have:
\[ \frac{du}{dt} = 28t^3 - 6t^2 - 8 \]

### Step 2: Evaluate \( \frac{du}{dt} \) at \( t = 0 \)
Substituting \( t = 0 \) into the derivative:
\[ \frac{du}{dt} \bigg|_{t=0} = 28(0)^3 - 6(0)^2 - 8 = 0 - 0 - 8 = -8 \]

### Step 3: Evaluate \( \frac{du}{dt} \) at \( t = 1 \)
Substituting \( t = 1 \) into the derivative:
\[ \frac{du}{dt} \bigg|_{t=1} = 28(1)^3 - 6(1)^2 - 8 = 28 - 6 - 8 = 14 \]

### Step 4: Evaluate \( \frac{du}{dt} \) at \( t = 2 \)
Substituting \( t = 2 \) into the derivative:
\[ \frac{du}{dt} \bigg|_{t=2} = 28(2)^3 - 6(2)^2 - 8 = 28(8) - 6(4) - 8 = 224 - 24 - 8 = 192 \]

### Final Results
Thus, the values of \( \frac{du}{dt} \) at \( t = 0, 1, 2 \) are:
- At \( t = 0 \): \( \frac{du}{dt} = -8 \)
- At \( t = 1 \): \( \frac{du}{dt} = 14 \)
- At \( t = 2 \): \( \frac{du}{dt} = 192 \)