Proving that the product rule for differentiating products applies to vectors

In summary, the product rule for differentiating products is a formula used in calculus to find the derivative of a product of two functions. It also applies to vectors by finding the derivative of the product of two vector-valued functions. Proving that the product rule applies to vectors allows for easier and more efficient problem-solving. An example of using the product rule for differentiating products of vectors was provided. Other rules and properties, such as the chain rule and vector operations, also apply to differentiating vector-valued functions.
  • #1
a.merchant
4
0
If r and s are vectors that depend on time, prove that the product rule for differentiating products applies to r.s, that is that:

d/dt (r.s) = r. ds/dt + dr/dt .s


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I'm not entirely sure how I'm supposed to go about proving this, can anyone point me in the right direction, please?

Homework Statement

 
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Start with

[tex]\frac{d}{dt} (\vec{r}\cdot\vec{s}) = \frac{d}{dt}(r_x s_x + r_y s_y + r_z s_z)[/tex]
 

FAQ: Proving that the product rule for differentiating products applies to vectors

What is the product rule for differentiating products?

The product rule for differentiating products is a formula used in calculus to find the derivative of a product of two functions. It states that the derivative of the product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function. In mathematical notation, it can be written as (f*g)' = f'*g + f*g'.

How does the product rule apply to vectors?

The product rule also applies to vectors in a similar way. Instead of two functions, we have two vector-valued functions. The derivative of the product of two vector-valued functions is equal to the first function times the derivative of the second vector-valued function, plus the second vector-valued function times the derivative of the first function.

Why is it important to prove that the product rule applies to vectors?

Proving that the product rule applies to vectors is important because it allows us to use the same rule for differentiating products of functions in both scalar and vector calculus. This makes solving problems involving vector-valued functions much easier and more efficient.

Can you provide an example of how to use the product rule for differentiating products of vectors?

Yes, for example, if we have two vector-valued functions f(t) = (t^2, t) and g(t) = (2t, t^3), then the product of these two functions is (f*g)(t) = (2t^3, t^4). To find the derivative of this product, we use the product rule: (f*g)'(t) = f'(t)*g(t) + f(t)*g'(t) = (4t^2, 1)*(2t, t^3) + (t^2, t)*(2, 3t^2) = (8t^3, 2t^4) + (2t^3, 3t^3) = (10t^3, 5t^3).

Are there any other rules or properties that apply to differentiating vector-valued functions?

Yes, there are several other rules and properties that apply to differentiating vector-valued functions, such as the chain rule, the quotient rule, and the power rule. Additionally, the properties of vector operations, such as addition, subtraction, and scalar multiplication, also apply to differentiating vector-valued functions.

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