`s=5t^(3)-3t^(5)`

Options

A. `(ds)/(dt)=15t^(2)+15t^(4)`
B. `(ds)/(dt)=15t^(2)-5t^(4)`
C. `(ds)/(dt)=15t^(2)-15t^(4)`
D. `(ds)/(dt)=15t^(2)+5t^(4)`

Answer

To solve the problem, we need to differentiate the function \( s = 5t^3 - 3t^5 \) with respect to time \( t \). Let's go through the steps one by one.

### Step-by-Step Solution:

1. **Identify the function to differentiate**:
We have the function \( s = 5t^3 - 3t^5 \).

2. **Apply the differentiation rule**:
We will use the power rule of differentiation, which states that if \( f(t) = t^n \), then \( \frac{df}{dt} = n \cdot t^{n-1} \).

3. **Differentiate each term separately**:
We can differentiate the function term by term:
\[ \frac{ds}{dt} = \frac{d}{dt}(5t^3) - \frac{d}{dt}(3t^5) \]

4. **Differentiate the first term \( 5t^3 \)**:
Using the power rule:
\[ \frac{d}{dt}(5t^3) = 5 \cdot 3t^{3-1} = 15t^2 \]

5. **Differentiate the second term \( -3t^5 \)**:
Again using the power rule:
\[ \frac{d}{dt}(-3t^5) = -3 \cdot 5t^{5-1} = -15t^4 \]

6. **Combine the results**:
Now we combine the derivatives of both terms:
\[ \frac{ds}{dt} = 15t^2 - 15t^4 \]

7. **Final result**:
Thus, the derivative of \( s \) with respect to \( t \) is:
\[ \frac{ds}{dt} = 15t^2 - 15t^4 \]

### Final Answer:
\[ \frac{ds}{dt} = 15t^2 - 15t^4 \]