https://file.aiquickdraw.com/mathbot/file/1725928455962axa0222y.jpg

We are given the vector function:

\[
r(t) = -6t^8\mathbf{i} + 9\cos(t)\mathbf{j} + e^{5t}\mathbf{k}
\]

We need to compute the derivative of \(3r(t)\), that is:

\[
\frac{d}{dt}\left[3r(t)\right]
\]

### Step 1: Multiply the vector \(r(t)\) by 3
We first compute \(3r(t)\):

\[
3r(t) = 3 \left( -6t^8\mathbf{i} + 9\cos(t)\mathbf{j} + e^{5t}\mathbf{k} \right)
= -18t^8\mathbf{i} + 27\cos(t)\mathbf{j} + 3e^{5t}\mathbf{k}
\]

### Step 2: Differentiate each component with respect to \(t\)

#### Derivative of the \(i\)-component: 
\[
\frac{d}{dt}(-18t^8) = -18 \cdot 8t^7 = -144t^7
\]

#### Derivative of the \(j\)-component: 
\[
\frac{d}{dt}(27\cos(t)) = -27\sin(t)
\]

#### Derivative of the \(k\)-component: 
\[
\frac{d}{dt}(3e^{5t}) = 3 \cdot 5e^{5t} = 15e^{5t}
\]

### Step 3: Combine the results

\[
\frac{d}{dt}\left[3r(t)\right] = -144t^7\mathbf{i} - 27\sin(t)\mathbf{j} + 15e^{5t}\mathbf{k}
\]

Thus, the derivative is:

\[
\boxed{-144t^7\mathbf{i} - 27\sin(t)\mathbf{j} + 15e^{5t}\mathbf{k}}
\]

Let me know if you'd like further explanation or have questions!

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Here are 5 related questions that might help expand your understanding:
1. How would the process change if the function had a time-dependent scalar in front of \(r(t)\)?
2. What is the geometric meaning of the derivative of a vector-valued function?
3. How do you differentiate other trigonometric functions like \(\sin(t)\) or \(\tan(t)\) in a vector context?
4. What is the second derivative \(\frac{d^2r(t)}{dt^2}\) for this vector function?
5. How does this differentiation process extend to applications in physics, like velocity and acceleration?

**Tip:** When differentiating vector functions, treat each component separately, as you would in regular differentiation. The rules for derivatives (like product rule, chain rule) still apply component-wise.

Given r(t) = -6t^8i + 9cos(t)j + e^5tk, find d/dt [3r(t)] at any t.

Find the derivative of the vector function r(t) = -6t^8i + 9cos(t)j + e^5tk. This problem explores key concepts in vector calculus, including differentiation of scalar functions and vector components, along with the chain and product rules.

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