What is Chain rule: Definition and 508 Discussions
In calculus, the chain rule is a formula to compute the derivative of a composite function. That is, if f and g are differentiable functions, then the chain rule expresses the derivative of their composite f ∘ g — the function which maps x to
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{\displaystyle f(g(x))}
— in terms of the derivatives of f and g and the product of functions as follows:
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{\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.}
Alternatively, by letting h = f ∘ g (equiv., h(x) = f(g(x)) for all x), one can also write the chain rule in Lagrange's notation, as follows:
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{\displaystyle h'(x)=f'(g(x))g'(x).}
The chain rule may also be rewritten in Leibniz's notation in the following way. If a variable z depends on the variable y, which itself depends on the variable x (i.e., y and z are dependent variables), then z, via the intermediate variable of y, depends on x as well. In which case, the chain rule states that:
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{\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}.}
More precisely, to indicate the point each derivative is evaluated at,
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{\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=\left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x}}
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The versions of the chain rule in the Lagrange and the Leibniz notation are equivalent, in the sense that if
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{\displaystyle z=f(y)}
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{\displaystyle y=g(x)}
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{\displaystyle z=f(g(x))=(f\circ g)(x)}
, then
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{\displaystyle \left.{\frac {dz}{dx}}\right|_{x}=(f\circ g)'(x)}
and
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{\displaystyle \left.{\frac {dz}{dy}}\right|_{y(x)}\cdot \left.{\frac {dy}{dx}}\right|_{x}=f'(y(x))g'(x)=f'(g(x))g'(x).}
Intuitively, the chain rule states that knowing the instantaneous rate of change of z relative to y and that of y relative to x allows one to calculate the instantaneous rate of change of z relative to x. As put by George F. Simmons: "if a car travels twice as fast as a bicycle and the bicycle is four times as fast as a walking man, then the car travels 2 × 4 = 8 times as fast as the man."In integration, the counterpart to the chain rule is the substitution rule.
Hey what's up,
I had a question on the chain rule...How would I use the chain rule on a quotient...like if i have 1/(t^4 + 1)^3 , Would I use the quotient rule first, or just start with the chain rule?
Does anyone know how to do this with chain rule?
If a cone has height 1 m and radius 30 cm, and the height is increasing at a rate of 1 cm/s, whereas the radius is decreasing at a rate of 1 cm/s, what is the rate of change of the cones volume? Solve the problem using the chain rule...
Can someone give me hints on how to prove the chain rule? I want to be able to learn this.
Let h(x)=f(g(x)), then
h'(x)=\lim\limits_{h \to 0}\frac {f(g(x+h))-f(g(x))}{h}
But what now?
I'm having some trouble understanding why the chain rule comes into play in related rates problems. I'm able to follow the procedure outlined in the text and come up with the correct answer, but I'm not really comfortable moving on from a section until I understand why it works.
The...
I am having a hard time doing the following problems. First off all the notation is confusing the hell out of me. This is the first time i have used this notation so it is making learning very difficult. Here are my questions.
Prove the following function is differentiable, and find the...
I know in Logarithms loga b * logc d = loga d * logc b
and
loga b * logb c = loga c.
Chain Rule.
Now I read Calculus, I found out about the Chain rule, are they the same?? Looks like it. But because of my poor English reading, I couldn't understand the text. Can some one explain what...