Differentiation of multivariable function proofddx∫v(x)u(x)f(t,x)dt=u′(x)f(u(x),x)−v′(x)f(v(x),x)+∫v(x)u(x)∂∂xf(t,x)dt
Differentiation of multivariable function proof
Answer & Explanation
Beginner2022-01-06Added 38 answers
Start with Where F is the antiderivative of f Now for a function its derivative with respect to x can be written as In terms of F we have and which remember u and v are functions of x. Therefore for each of them we can write this by plugging into the above formula. or and . because Now the above also holds for so Therefore Now rearranging you can see that it starts to take on your form. Now the last two terms can be written in terms of an integral. Which then can all come together to give
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