48t + t^2 - (t^3/12) find the erivative and then simplify it 

To find the derivative of the expression \( 48t + t^2 - \frac{t^3}{12} \), we will follow the basic rules of differentiation.

### Step-by-Step Differentiation:
The expression is:

\[
f(t) = 48t + t^2 - \frac{t^3}{12}
\]

#### 1. Differentiate \( 48t \):
\[
\frac{d}{dt} (48t) = 48
\]

#### 2. Differentiate \( t^2 \):
\[
\frac{d}{dt} (t^2) = 2t
\]

#### 3. Differentiate \( \frac{t^3}{12} \):
We can apply the power rule, which says \( \frac{d}{dt} (t^n) = n t^{n-1} \), and multiply by the constant \( \frac{1}{12} \):

\[
\frac{d}{dt} \left( \frac{t^3}{12} \right) = \frac{1}{12} \cdot 3t^2 = \frac{t^2}{4}
\]

### Combine the results:
Now, we combine all the derivatives:

\[
f'(t) = 48 + 2t - \frac{t^2}{4}
\]

### Simplify the expression:
We can leave it in this form, or if needed, find a common denominator to simplify further.

1. The least common denominator is 4, so express all terms with denominator 4:

\[
f'(t) = \frac{192}{4} + \frac{8t}{4} - \frac{t^2}{4}
\]

2. Combine the terms:

\[
f'(t) = \frac{192 + 8t - t^2}{4}
\]

Thus, the simplified derivative is:

\[
f'(t) = \frac{192 + 8t - t^2}{4}
\]

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you apply the power rule of differentiation to negative exponents?
2. How would the second derivative of \( f(t) \) look, and what does it represent?
3. What is the significance of the constant \( \frac{1}{12} \) when differentiating the term \( \frac{t^3}{12} \)?
4. Can this expression \( f(t) \) have critical points? How would you find them?
5. How can we apply this differentiation to a real-world problem?

**Tip**: Always check for opportunities to factor or simplify an expression further after differentiation.

48t + t^2 - (t^3/12) find the derivative and then simplify it

Learn how to find and simplify the derivative of the expression 48t + t^2 - (t^3/12). This step-by-step solution explains key differentiation rules, including the power rule and constant multiple rule. Suitable for students in Grades 11-12 studying calculus or advanced algebra.

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