# Can you explain how the chain rule work in real life?

See below.

This is the best analogy I can think of.

Person a

Person b

Person c

If person b walks twice as fast as person c, we could represent this as:

If person a walks twice as fast as person b, we could represent this as:

Now, how much faster than person c does person a walk.

Our intuition would be, if a is twice as fast as b , and b is twice as fast as c, then a is:

This in the chain rule is:

.i.e.

Hopefully from this you can see where the product of the rates of change come into play.

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The chain rule in calculus allows you to find the derivative of a composite function. In real life, it's analogous to situations where multiple processes or functions are intertwined. For example, in economics, the total cost of producing a product may involve several stages, each with its own rate of change. By using the chain rule, you can understand how changes in one stage affect the overall cost. Similarly, in physics, when studying motion through different mediums with varying properties, the chain rule helps determine how changes in one variable affect the overall motion. In essence, the chain rule helps analyze complex systems by breaking them down into smaller, more manageable components.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the implicit derivative of #1= e^y-xcos(xy) #?
- How do you differentiate #f(x)= (1 - sin^2x)/(x-cosx) # using the quotient rule?
- How do you differentiate #f(x) = sin(xcos(x))# using the chain rule?
- What is the second derivative of #x^2 + (16/x)#?
- How do you differentiate #g(u) =(u^-2 + u^3)(u^5 - u^-2) # using the product rule?

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