Let 
\[h\] be a vector-valued function defined by 
 
\[h(t)=\left(-\dfrac{3}{t+2},e^{3t}\right)\].
Find 
\[h\]'s second derivative 
\[h''(t)\].

Given the vector-valued function \( h(t) = \left(-\frac{3}{t+2}, e^{3t}\right) \), we need to find the second derivative \( h''(t) \).

### Step 1: Find the first derivative \( h'(t) \).

Let's differentiate each component of \( h(t) \) with respect to \( t \).

1. **Differentiate the first component:**
   \[
   \frac{d}{dt}\left(-\frac{3}{t+2}\right)
   \]
   This is a quotient rule problem where:
   \[
   \text{Numerator } (u) = -3, \quad \text{Denominator } (v) = t + 2
   \]
   Applying the quotient rule:
   \[
   \frac{d}{dt}\left(-\frac{3}{t+2}\right) = \frac{(0)(t+2) - (-3)(1)}{(t+2)^2} = \frac{3}{(t+2)^2}
   \]

2. **Differentiate the second component:**
   \[
   \frac{d}{dt}(e^{3t}) = 3e^{3t}
   \]

Thus, the first derivative \( h'(t) \) is:
\[
h'(t) = \left(\frac{3}{(t+2)^2}, 3e^{3t}\right)
\]

### Step 2: Find the second derivative \( h''(t) \).

Now, differentiate each component of \( h'(t) \) with respect to \( t \) again.

1. **Differentiate the first component of \( h'(t) \):**
   \[
   \frac{d}{dt}\left(\frac{3}{(t+2)^2}\right)
   \]
   This can be simplified as:
   \[
   \frac{3}{(t+2)^2} = 3(t+2)^{-2}
   \]
   Now differentiate:
   \[
   \frac{d}{dt} \left[ 3(t+2)^{-2} \right] = 3 \cdot (-2) \cdot (t+2)^{-3} \cdot 1 = -\frac{6}{(t+2)^3}
   \]

2. **Differentiate the second component of \( h'(t) \):**
   \[
   \frac{d}{dt}(3e^{3t}) = 9e^{3t}
   \]

Thus, the second derivative \( h''(t) \) is:
\[
h''(t) = \left(-\frac{6}{(t+2)^3}, 9e^{3t}\right)
\]

Would you like any further details or explanations?

### Related Questions:

1. How do you apply the chain rule to vector-valued functions?
2. What is the geometric interpretation of a vector-valued function’s second derivative?
3. Can you find the first derivative of the magnitude of \( h(t) \)?
4. How does the quotient rule apply in more complex functions?
5. What are some examples of vector-valued functions used in physics?

**Tip:** When differentiating vector-valued functions, treat each component as a separate function, applying the relevant rules accordingly.

Learn how to find the second derivative of the vector-valued function h(t) = (-3/(t+2), e^(3t)). Step-by-step differentiation process explained with the application of quotient rule and exponential differentiation. Suitable for advanced high school and college students.

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