Proving Leibnitz rule for integration. [closed]

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Let $I=[a,b] subset mathbb R$ and let $f : I longrightarrow mathbb R$ be a continuous function on $I$. Let $J=[c,d] subset mathbb R$ and let $u : J longrightarrow mathbb R$ be differentiable on $J$ and $u(J) subset I$; $v : J longrightarrow mathbb R$ be differentiable on $J$ and $v(J) subset I$. If $g : J longrightarrow mathbb R$ be defined by $g(t)=int_u(t)^v(t) f(x) dx$ for $t in J$, then $g'(t) = (f circ v)(t).v'(t) - (f circ u)(t).u'(t)$ for all $t in J$.





How do I prove it? Please help me.



Thank you very much.






real-analysis riemann-integration







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closed as off-topic by Nosrati, Paramanand Singh, Theoretical Economist, Jendrik Stelzner, José Carlos Santos Sep 7 at 15:32


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Paramanand Singh, Theoretical Economist, Jendrik Stelzner, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.












  • Use definition of derivatives.
    – xbh
    Aug 31 at 3:21










  • Look at Papa Flammy's proof
    – John Glenn
    Aug 31 at 3:29










  • Start by showing the derivative of $int_c^t f (x) dx$ is $f(t)$.
    – copper.hat
    Aug 31 at 3:46










  • Do you know fundamental theorem of calculus? Use it together with chain rule of derivatives and you are done.
    – Paramanand Singh
    Aug 31 at 4:27














up vote
0
down vote

favorite














Let $I=[a,b] subset mathbb R$ and let $f : I longrightarrow mathbb R$ be a continuous function on $I$. Let $J=[c,d] subset mathbb R$ and let $u : J longrightarrow mathbb R$ be differentiable on $J$ and $u(J) subset I$; $v : J longrightarrow mathbb R$ be differentiable on $J$ and $v(J) subset I$. If $g : J longrightarrow mathbb R$ be defined by $g(t)=int_u(t)^v(t) f(x) dx$ for $t in J$, then $g'(t) = (f circ v)(t).v'(t) - (f circ u)(t).u'(t)$ for all $t in J$.





How do I prove it? Please help me.



Thank you very much.






real-analysis riemann-integration







share|cite|improve this question













closed as off-topic by Nosrati, Paramanand Singh, Theoretical Economist, Jendrik Stelzner, José Carlos Santos Sep 7 at 15:32


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Paramanand Singh, Theoretical Economist, Jendrik Stelzner, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.












  • Use definition of derivatives.
    – xbh
    Aug 31 at 3:21










  • Look at Papa Flammy's proof
    – John Glenn
    Aug 31 at 3:29










  • Start by showing the derivative of $int_c^t f (x) dx$ is $f(t)$.
    – copper.hat
    Aug 31 at 3:46










  • Do you know fundamental theorem of calculus? Use it together with chain rule of derivatives and you are done.
    – Paramanand Singh
    Aug 31 at 4:27












up vote
0
down vote

favorite









up vote
0
down vote

favorite













Let $I=[a,b] subset mathbb R$ and let $f : I longrightarrow mathbb R$ be a continuous function on $I$. Let $J=[c,d] subset mathbb R$ and let $u : J longrightarrow mathbb R$ be differentiable on $J$ and $u(J) subset I$; $v : J longrightarrow mathbb R$ be differentiable on $J$ and $v(J) subset I$. If $g : J longrightarrow mathbb R$ be defined by $g(t)=int_u(t)^v(t) f(x) dx$ for $t in J$, then $g'(t) = (f circ v)(t).v'(t) - (f circ u)(t).u'(t)$ for all $t in J$.





How do I prove it? Please help me.



Thank you very much.






real-analysis riemann-integration







share|cite|improve this question















Let $I=[a,b] subset mathbb R$ and let $f : I longrightarrow mathbb R$ be a continuous function on $I$. Let $J=[c,d] subset mathbb R$ and let $u : J longrightarrow mathbb R$ be differentiable on $J$ and $u(J) subset I$; $v : J longrightarrow mathbb R$ be differentiable on $J$ and $v(J) subset I$. If $g : J longrightarrow mathbb R$ be defined by $g(t)=int_u(t)^v(t) f(x) dx$ for $t in J$, then $g'(t) = (f circ v)(t).v'(t) - (f circ u)(t).u'(t)$ for all $t in J$.





How do I prove it? Please help me.



Thank you very much.






real-analysis riemann-integration




real-analysis riemann-integration






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




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asked Aug 31 at 3:10









Dbchatto67

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closed as off-topic by Nosrati, Paramanand Singh, Theoretical Economist, Jendrik Stelzner, José Carlos Santos Sep 7 at 15:32


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Paramanand Singh, Theoretical Economist, Jendrik Stelzner, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by Nosrati, Paramanand Singh, Theoretical Economist, Jendrik Stelzner, José Carlos Santos Sep 7 at 15:32


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nosrati, Paramanand Singh, Theoretical Economist, Jendrik Stelzner, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.











  • Use definition of derivatives.
    – xbh
    Aug 31 at 3:21










  • Look at Papa Flammy's proof
    – John Glenn
    Aug 31 at 3:29










  • Start by showing the derivative of $int_c^t f (x) dx$ is $f(t)$.
    – copper.hat
    Aug 31 at 3:46










  • Do you know fundamental theorem of calculus? Use it together with chain rule of derivatives and you are done.
    – Paramanand Singh
    Aug 31 at 4:27
















  • Use definition of derivatives.
    – xbh
    Aug 31 at 3:21










  • Look at Papa Flammy's proof
    – John Glenn
    Aug 31 at 3:29










  • Start by showing the derivative of $int_c^t f (x) dx$ is $f(t)$.
    – copper.hat
    Aug 31 at 3:46










  • Do you know fundamental theorem of calculus? Use it together with chain rule of derivatives and you are done.
    – Paramanand Singh
    Aug 31 at 4:27















Use definition of derivatives.
– xbh
Aug 31 at 3:21




Use definition of derivatives.
– xbh
Aug 31 at 3:21












Look at Papa Flammy's proof
– John Glenn
Aug 31 at 3:29




Look at Papa Flammy's proof
– John Glenn
Aug 31 at 3:29












Start by showing the derivative of $int_c^t f (x) dx$ is $f(t)$.
– copper.hat
Aug 31 at 3:46




Start by showing the derivative of $int_c^t f (x) dx$ is $f(t)$.
– copper.hat
Aug 31 at 3:46












Do you know fundamental theorem of calculus? Use it together with chain rule of derivatives and you are done.
– Paramanand Singh
Aug 31 at 4:27




Do you know fundamental theorem of calculus? Use it together with chain rule of derivatives and you are done.
– Paramanand Singh
Aug 31 at 4:27















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