A body is moving according to the equation `x=at+bt^(2)-ct^(3)`. Then its instantaneous speed is given By :-

Explanation

To find the instantaneous speed of a body moving according to the equation \( x = at + bt^2 - ct^3 \), we need to differentiate the displacement equation with respect to time \( t \). Here’s a step-by-step solution:

### Step 1: Write down the displacement equation
The displacement of the body is given by:
\[ x = at + bt^2 - ct^3 \]

### Step 2: Differentiate the displacement equation
To find the instantaneous speed (or instantaneous velocity), we differentiate \( x \) with respect to \( t \):
\[ \frac{dx}{dt} = \frac{d}{dt}(at + bt^2 - ct^3) \]

### Step 3: Apply the differentiation rules
Using the power rule of differentiation:
- The derivative of \( at \) is \( a \).
- The derivative of \( bt^2 \) is \( 2bt \).
- The derivative of \( -ct^3 \) is \( -3ct^2 \).

Putting it all together, we have:
\[ \frac{dx}{dt} = a + 2bt - 3ct^2 \]

### Step 4: Write the final expression for instantaneous speed
Thus, the instantaneous speed of the body is given by:
\[ \text{Instantaneous Speed} = a + 2bt - 3ct^2 \]

### Summary
The instantaneous speed of the body moving according to the equation \( x = at + bt^2 - ct^3 \) is:
\[ \text{Instantaneous Speed} = a + 2bt - 3ct^2 \]

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