Problem 50 - Thomas Calculus

Question

Sum and Difference Rules Prove that if \(\mathbf{u}\) and \(\mathbf{v}\) are differentiable functions of \(t,\) then $$ \frac{d}{d t}(\mathbf{u}+\mathbf{v})=\frac{d \mathbf{u}}{d t}+\frac{d \mathbf{v}}{d t} $$ and $$ \frac{d}{d t}(\mathbf{u}-\mathbf{v})=\frac{d \mathbf{u}}{d t}-\frac{d \mathbf{v}}{d t} $$

Step-by-Step Solution

Answer

The derivative rules for sums and differences are proven using the limit definition.

Step 1: Definition of Derivative

The derivative of a function with respect to a variable measures how the function changes as that variable changes. For a function \( \mathbf{u}(t) \), the derivative \( \frac{d \mathbf{u}}{d t} \) represents the rate of change of \( \mathbf{u} \) with respect to \( t \).

Step 2: Sum of Functions

Consider the function \( \mathbf{w}(t) = \mathbf{u}(t) + \mathbf{v}(t) \). The derivative of \( \mathbf{w} \) with respect to \( t \) is given by:\[ \frac{d}{d t} (\mathbf{u} + \mathbf{v}) = \lim_{\Delta t \to 0} \frac{(\mathbf{u}(t + \Delta t) + \mathbf{v}(t + \Delta t)) - (\mathbf{u}(t) + \mathbf{v}(t))}{\Delta t} \]

Step 3: Apply the Limit Property

Using the limit property that \( \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \), rewrite the expression:\[ \frac{d}{d t} (\mathbf{u} + \mathbf{v}) = \lim_{\Delta t \to 0} \left( \frac{\mathbf{u}(t + \Delta t) - \mathbf{u}(t)}{\Delta t} + \frac{\mathbf{v}(t + \Delta t) - \mathbf{v}(t)}{\Delta t} \right) \]

Step 4: Separating Limits

Separate the limits of the numerator's differences over \( \Delta t \):\[ \frac{d}{d t} (\mathbf{u} + \mathbf{v}) = \left( \lim_{\Delta t \to 0} \frac{\mathbf{u}(t + \Delta t) - \mathbf{u}(t)}{\Delta t} \right) + \left( \lim_{\Delta t \to 0} \frac{\mathbf{v}(t + \Delta t) - \mathbf{v}(t)}{\Delta t} \right) \]

Step 5: Recognizing Individual Derivatives

Recognize that the separated limits are simply the derivatives of \( \mathbf{u} \) and \( \mathbf{v} \) respectively. Thus:\[ \frac{d}{d t} (\mathbf{u} + \mathbf{v}) = \frac{d \mathbf{u}}{d t} + \frac{d \mathbf{v}}{d t} \]

Step 6: Difference of Functions

Consider the function \( \mathbf{w}(t) = \mathbf{u}(t) - \mathbf{v}(t) \). The derivative of \( \mathbf{w} \) with respect to \( t \) follows a similar pattern as shown in the sum:\[ \frac{d}{d t} (\mathbf{u} - \mathbf{v}) = \lim_{\Delta t \to 0} \frac{(\mathbf{u}(t + \Delta t) - \mathbf{v}(t + \Delta t)) - (\mathbf{u}(t) - \mathbf{v}(t))}{\Delta t} \]

Step 7: Apply Limit for Difference Rule

Use the same limit property focusing on subtraction:\[ \frac{d}{d t} (\mathbf{u} - \mathbf{v}) = \left( \lim_{\Delta t \to 0} \frac{\mathbf{u}(t + \Delta t) - \mathbf{u}(t)}{\Delta t} \right) - \left( \lim_{\Delta t \to 0} \frac{\mathbf{v}(t + \Delta t) - \mathbf{v}(t)}{\Delta t} \right) \]

Step 8: Recognizing Individual Derivatives Again

The separated limits are again the derivatives of \( \mathbf{u} \) and \( \mathbf{v} \) respectively. Hence:\[ \frac{d}{d t} (\mathbf{u} - \mathbf{v}) = \frac{d \mathbf{u}}{d t} - \frac{d \mathbf{v}}{d t} \]

Step 9: Conclusion

We have shown that the derivative of the sum of two functions is the sum of their derivatives, and the derivative of the difference is the difference of their derivatives, proving the sum and difference rules.

Key Concepts

Sum Rule

When dealing with derivatives, one of the first rules you encounter is the Sum Rule. In simple terms, this rule tells us how to differentiate the sum of two functions. Let's say you have two functions \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \), both of which are differentiable. According to the Sum Rule, if you want to find the derivative of \( \mathbf{u}(t) + \mathbf{v}(t) \), you simply take the derivative of each function individually and then add these derivatives together.

So, mathematically speaking, if \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \) are functions of \( t \), the rule is expressed as:

• \( \frac{d}{dt} (\mathbf{u} + \mathbf{v}) = \frac{d \mathbf{u}}{dt} + \frac{d \mathbf{v}}{dt} \)

This can be visualized as breaking down a complex task into simpler, more manageable tasks. You differentiate each component function and then combine your results, making it a straightforward way to handle the sum of functions.

Difference Rule

Closely related to the Sum Rule is the Difference Rule. This rule addresses how to differentiate the difference between two functions. It's almost like a mirror image of the Sum Rule, but instead of adding the derivatives, we subtract them.

Imagine you have two functions \( \mathbf{u}(t) \) and \( \mathbf{v}(t) \) that are differentiable. To find the derivative of their difference \( \mathbf{u}(t) - \mathbf{v}(t) \), the Difference Rule tells us to take the derivative of each function separately and subtract one from the other.

Here’s how it looks mathematically:

• \( \frac{d}{dt} (\mathbf{u} - \mathbf{v}) = \frac{d \mathbf{u}}{dt} - \frac{d \mathbf{v}}{dt} \)

By implementing the Difference Rule, you handle subtractions much like additions, only instead of summing up changes, you're calculating the difference in changes between the two functions.

Derivative of a Function

The derivative of a function is a core concept in calculus and serves as a measure of how a function changes as its input changes. In essence, it tells you the rate at which the function's value is changing at any given point.

Suppose you have a function \( \mathbf{u}(t) \); its derivative, denoted by \( \frac{d \mathbf{u}}{dt} \), indicates how \( \mathbf{u} \) changes with respect to \( t \). This conceptual tool is crucial when analyzing any dynamic system where variables are in continual flux.

The act of finding a derivative is known as differentiation, and it uses specific rules like the Sum Rule and Difference Rule to systematically determine these rates of change.

Limit Definition of the Derivative

The fundamental way to define a derivative is through the limit process. This definition gives the derivative its formal mathematical meaning and illustrates how small changes in a function's input lead to changes in output.

For a function \( \mathbf{u}(t) \), the derivative \( \frac{d \mathbf{u}}{dt} \) is defined as:

• \( \lim_{\Delta t \to 0} \frac{\mathbf{u}(t + \Delta t) - \mathbf{u}(t)}{\Delta t} \)

This means you take the difference between the values of the function at two points very close together, divide by the change in the input \( \Delta t \), and see what value this quotient approaches as \( \Delta t \) becomes infinitely small.

Limit definition is not just a formula but a concept that underpins the derivative. It provides an essential understanding that enhances the comprehension of more advanced calculus topics.