Let 𝐮(t)=2 t^3 𝐢+(t^2-1) 𝐣-8 𝐤 and 𝐯(t)=e^t 𝐢+2 e^-t 𝐣-e^2...

Let $$\mathbf{u}(t)=2 t^{3} \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}-8 \mathbf{k} \text { and } \mathbf{v}(t)=e^{t} \mathbf{i}+2 e^{-t} \mathbf{j}-e^{2 t} \mathbf{k}$$ Compute the derivative of the following functions.
$$\mathbf{v}(\sqrt{t})$$

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Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus that allows you to differentiate composite functions. When a function is composed of an inner function and an outer function, the derivative of the composite function is obtained by multiplying the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function. This concept is crucial when dealing with functions that are composed of other functions, as seen in many practical applications including vector-valued functions.

Differentiation of Vector-Valued Functions

Differentiation of vector-valued functions extends the concept of derivative to functions that output vectors instead of scalars. Each component of the vector is differentiated separately, and the resulting derivative is a vector composed of these individual derivatives. This process is essential in multivariable calculus and physics, where quantities like position, velocity, and acceleration are naturally represented as vectors.

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