Proof of distributive and product rule

In summary, the standard product rule states that (fg)'=f'g+fg'. This proof should be relatively easy to follow if you're familiar with limits and exponents.
  • #1
srhelfrich
7
0
1. Prove a) r=(u*v)=r*u+r*v and b) d/dt(r*s)=r*ds/st+dr/dt*s
2. Homework Equations : b) dr/dt=lim t->0=Δr/Δt and Δr=r(t+Δt)-r(t)
3. Attempt at the solution:
Okay, so I was able to work out part a but I'm not quite sure how to start part b. Could anyone point me toward a useful resource to explain how to approach this problem? I'm not looking for an answer, just a means to an answer.

(* is dot product)
 
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  • #2
srhelfrich said:
1. Prove a) r=(u*v)=r*u+r*v and b) d/dt(r*s)=r*ds/st+dr/dt*s



2. Homework Equations : b) dr/dt=lim t->0=Δr/Δt and Δr=r(t+Δt)-r(t)



3. Attempt at the solution:
Okay, so I was able to work out part a but I'm not quite sure how to start part b. Could anyone point me toward a useful resource to explain how to approach this problem? I'm not looking for an answer, just a means to an answer.

(* is dot product)

Are you familiar with a proof of the standard product rule; i.e. the version that says (fg)'=f'g+fg'? The proof of this version shouldn't look significantly different.
 
  • #3
Thank you! I looked it up and you're right, I see where they start with the limit and expand from there. Much appreciated!
 
  • #4
Alright then. The key things you'll want to verify (if you haven't already) are that the algebra $$\mathbf{u}\cdot(\mathbf{v}+\mathbf{w})=\mathbf u\cdot\mathbf v+\mathbf{u}\cdot\mathbf{w},\ \ c(\mathbf{u}\cdot\mathbf{v})=(c\mathbf{u})\cdot\mathbf{v}$$ and calculus
$$\lim_{t\rightarrow a}\left(\mathbf u(t)+\mathbf v(t)\right)=\lim_{t\rightarrow a}\mathbf u(t)+\lim_{t\rightarrow a}\mathbf v(t), \ \ \lim_{t\rightarrow a}\left(\mathbf u(t)\cdot\mathbf v(t)\right)=\lim_{t\rightarrow a}\mathbf u(t)\cdot\lim_{t\rightarrow a}\mathbf v(t)$$
work the same with vectors/vector-valued functions as they do with numbers/real-valued functions.
 
  • #5
srhelfrich said:
1. Prove a) r=(u*v)=r*u+r*v

This certainly isn't true! You mean "r*(u+ v)= r*u+ r*v" don't you?

and b) d/dt(r*s)=r*ds/st+dr/dt*s
2. Homework Equations : b) dr/dt=lim t->0=Δr/Δt and Δr=r(t+Δt)-r(t)



3. Attempt at the solution:
Okay, so I was able to work out part a but I'm not quite sure how to start part b. Could anyone point me toward a useful resource to explain how to approach this problem? I'm not looking for an answer, just a means to an answer.

(* is dot product)
 

1. What is the distributive property?

The distributive property is a mathematical rule that states that when multiplying a number by a sum or difference, the result is the same as multiplying each addend or subtrahend by the number and then adding or subtracting the products. In algebra, it is commonly written as a(b+c) = ab + ac.

2. How does the distributive property apply to variables?

The distributive property also applies to variables in the same way it applies to numbers. For example, in the expression 3(x+y), the 3 is distributed to both the x and y, resulting in 3x + 3y. This can also be applied to more complex expressions with multiple variables.

3. What is the product rule in calculus?

The product rule is a calculus rule used to find the derivative of a product of two functions. It states that the derivative of a product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.

4. How is the product rule different from the power rule?

The power rule is used to find the derivative of a function raised to a constant power, while the product rule is used to find the derivative of a product of two functions. The power rule is a special case of the product rule, where one of the functions is a constant.

5. Why is it important to understand the distributive and product rule?

The distributive and product rule are fundamental concepts in algebra and calculus, which are used extensively in higher level mathematics and in many scientific fields. They allow for simplification and manipulation of complex expressions, making problem-solving easier and more efficient. Understanding these rules is crucial for success in advanced mathematics and scientific research.

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