Math Problem Statement

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

Solution

We are given the vector function:

r(t)=6t8i+9cos(t)j+e5tkr(t) = -6t^8\mathbf{i} + 9\cos(t)\mathbf{j} + e^{5t}\mathbf{k}

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

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

Step 1: Multiply the vector r(t)r(t) by 3

We first compute 3r(t)3r(t):

= -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}}$$ Let me know if you'd like further explanation or have questions! --- Here are 5 related questions that might help expand your understanding: 1. How would the process change if the function had a time-dependent scalar in front of $$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.

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Math Problem Analysis

Mathematical Concepts

Vector Calculus
Differentiation

Formulas

Derivative of a scalar multiplied by a vector: d/dt [c * r(t)] = c * d/dt [r(t)]
Derivative of a trigonometric function: d/dt[cos(t)] = -sin(t)
Derivative of an exponential function: d/dt[e^(at)] = a * e^(at)

Theorems

Basic rules of vector differentiation
Chain rule
Product rule

Suitable Grade Level

Undergraduate