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Differentiation is a fundamental concept in calculus that involves finding the rate at which a function is changing. In the context of vector-valued functions, differentiation requires us to apply the derivative to each component of the vector function independently. A vector-valued function like \( \mathbf{r}(t) = e^{-t} \mathbf{i} + \sin(3t) \mathbf{j} + 10\sqrt{t} \mathbf{k} \) is composed of scalar functions that correspond to the \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) directions. We differentiate these scalar functions individually to determine the overall derivative \( \mathbf{r}'(t) \).

• The differentiation process respects linearity, meaning each term in the vector is differentiated separately.
• Understanding each component independently is crucial for computing derivatives of vector-valued functions.

The chain rule is a key differentiation technique used whenever a function is composed of other functions, i.e., when you need to differentiate a "function within a function." It is particularly useful in vector-valued functions where one of the components might involve a composite function. For example, in our function \( \mathbf{r}(t) = e^{-t} \mathbf{i} + \sin(3t) \mathbf{j} + 10\sqrt{t} \mathbf{k} \), the sine component \( \sin(3t) \) requires the chain rule.
To apply the chain rule:

• First, differentiate the outer function.
• Then multiply by the derivative of the inner function.

For \( \sin(3t) \), start by differentiating \( \sin(u) \) with respect to \( u \), which is \( \cos(u) \). Then differentiate \( 3t \) with respect to \( t \), resulting in \( 3 \). The product of these gives the derivative: \( 3\cos(3t) \). This allows us to handle components that might otherwise seem complex.

The exponential function is another important element frequently encountered in calculus, especially when differentiating vector-valued functions. The defining property of an exponential function is that it grows continuously and can be represented as \( e^x \), where \( e \) is Euler's number. A typical feature is its unique derivative property: the derivative of \( e^x \) is simply \( e^x \) itself.
For functions like \( e^{-t} \), the chain rule once again steps in. The direct differentiation gives us \( e^{-t} \), but account for the \( -t \) term by multiplying by the derivative of \( -t \), which is \( -1 \). Consequently, \( \frac{d}{dt}(e^{-t}) = -e^{-t} \).

• Remember, for \( e^x \), derivative equals itself, but adjust for any coefficients in the exponent with the chain rule.
• In vector contexts, it's vital to view each direction (\( \mathbf{i}, \mathbf{j}, \mathbf{k} \)) separately for clarity.

The power rule is an essential tool in differentiation used for functions of the form \( x^n \). This rule states that the derivative of \( x^n \) with respect to \( x \) is \( nx^{n-1} \). It's particularly helpful when dealing with components that involve polynomials or roots within vector-valued functions.
To differentiate expressions like \( 10\sqrt{t} \), you first reimagine the square root using exponential notation \( t^{1/2} \). Applying the power rule here gives us \( 10 \times \frac{1}{2}t^{-1/2} = 5t^{-1/2} \). This makes differentiating even complex polynomials straightforward.

• Adjust your exponent notation for square roots and other roots as fractions before differentiating.
• Always multiply by constants outside the function as shown in the example.
• Simplify the expression after differentiating to keep the derivative clean.