find the first and second derivative of u_p=Atcos(4sqrt3)+Btsin(4sqrt3)

To find the first and second derivatives of the function 

\[
u_p = A t \cos(4\sqrt{3}) + B t \sin(4\sqrt{3}),
\]

we'll denote \( A \) and \( B \) as constants and differentiate with respect to \( t \).

### Step 1: First Derivative \( u_p' \)
Using the product rule, \(\frac{d}{dt}(f(t)g(t)) = f'(t)g(t) + f(t)g'(t)\), we differentiate each term separately.

#### Differentiate \( A t \cos(4\sqrt{3}) \):
1. Since \( A \) and \( \cos(4\sqrt{3}) \) are constants, 
   \[
   \frac{d}{dt} (A t \cos(4\sqrt{3})) = A \cos(4\sqrt{3}).
   \]

#### Differentiate \( B t \sin(4\sqrt{3}) \):
2. Similarly, \( B \) and \( \sin(4\sqrt{3}) \) are constants, so
   \[
   \frac{d}{dt} (B t \sin(4\sqrt{3})) = B \sin(4\sqrt{3}).
   \]

Therefore, the first derivative \( u_p' \) is:
\[
u_p' = A \cos(4\sqrt{3}) + B \sin(4\sqrt{3}).
\]

### Step 2: Second Derivative \( u_p'' \)
Since \( u_p' \) is constant, the second derivative \( u_p'' \) will simply be zero:
\[
u_p'' = 0.
\]

### Final Answer
1. **First Derivative**: \( u_p' = A \cos(4\sqrt{3}) + B \sin(4\sqrt{3}) \)
2. **Second Derivative**: \( u_p'' = 0 \)

Would you like more details on the steps or have any questions?

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#### Related Questions
1. What happens to the derivative if \( \cos(4\sqrt{3}) \) or \( \sin(4\sqrt{3}) \) varies with \( t \)?
2. How would the result change if the function had an additional term with \( t^2 \) in it?
3. Why does a linear term in \( t \) produce a constant when differentiated?
4. How can product and chain rules apply in more complex derivatives of similar forms?
5. How does this differentiation apply in real-world contexts, like physics problems?

#### Tip
Always simplify constant terms after differentiation to avoid unnecessary computations in further steps.

Find the first and second derivative of u_p = A t cos(4 sqrt 3) + B t sin(4 sqrt 3).

Learn to find the first and second derivatives of the function u_p = Atcos(4√3) + Btsin(4√3) using differentiation techniques. This step-by-step solution applies the product rule and explores the derivative of a constant, providing insights for high school calculus students.

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