How Does the Chain Rule Apply to Finding Antiderivatives in the Volume Equation?

In summary, the conversation discusses the process of finding an antiderivative with respect to t for the equation (v^-.5)v'=k, where v is a function of t. The use of the chain rule and elementary antiderivative formulas is mentioned, and the individual is unsure of how to apply the chain rule in this situation. They also question if they should expand v' to v'*dv/dt. The expert confirms that v' is equivalent to dv/dt and suggests that the antiderivative should be 2(v^.5)=kt+C. They also provide a tip to check the answer by differentiating it.
  • #1
caljuice
70
0
The text is giving me steps on how to do it but I don't get what's it is asking.

The equation is (v^-.5)v'=k where v= volume

Then says we consider the chain rule together with elementary antiderivative formulas to determine an antiderivative with respect to t of (v^-.5)v'

I haven't learned integrals yet. I'm exactly sure I'm even looking for. I don't really need an answer but explanation how chain rule can applies here. Since V is a function of t, do I expand v' to v'*dv/dt?
 
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  • #2
caljuice said:
The text is giving me steps on how to do it but I don't get what's it is asking.

The equation is (v^-.5)v'=k where v= volume

Then says we consider the chain rule together with elementary antiderivative formulas to determine an antiderivative with respect to t of (v^-.5)v'

I haven't learned integrals yet. I'm exactly sure I'm even looking for. I don't really need an answer but explanation how chain rule can applies here. Since V is a function of t, do I expand v' to v'*dv/dt?

v' already is dv/dt, remember u(v(t))' = u'(v(t))*v'(t)
 
  • #3
ah okay thanks. So then the antiderivative should be

2(V^.5)=kt+C ?
 
  • #4
caljuice said:
ah okay thanks. So then the antiderivative should be

2(V^.5)=kt+C ?

That would be correct. If you are ever unsure, just differentiate your answer and see if you get what you started with.
 

Related to How Does the Chain Rule Apply to Finding Antiderivatives in the Volume Equation?

1. What is an antiderivative with chain rule?

An antiderivative with chain rule is a mathematical concept used to find the inverse of a derivative. It involves applying the chain rule, which allows for the differentiation of composite functions, to find the original function that was differentiated.

2. How is the chain rule applied in finding an antiderivative?

The chain rule is applied by first identifying the outer and inner functions of the composite function. The derivative of the outer function is then multiplied by the derivative of the inner function, which is found by applying the power rule or product rule. The resulting expression is then integrated to find the antiderivative.

3. Why is the antiderivative with chain rule important?

The antiderivative with chain rule is important because it allows for the reverse process of differentiation, which is useful in solving many real-world problems in fields such as physics, engineering, and economics. It also helps in finding the area under a curve, which is a fundamental concept in calculus.

4. What are some common examples of antiderivatives with chain rule?

Some common examples include finding the antiderivative of trigonometric functions, exponential and logarithmic functions, and polynomial functions. For example, the antiderivative of sinx would be -cosx, and the antiderivative of e^x would be e^x.

5. Are there any specific rules or techniques for solving antiderivatives with chain rule?

Yes, there are some specific rules and techniques that can be used to solve antiderivatives with chain rule. These include using substitution, integration by parts, and trigonometric identities. It is important to have a good understanding of these techniques and when to apply them in order to solve more complex antiderivatives.

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