Derivative of a multivariable function evaluate the derivative of a function

Stefan Hendricks

Stefan Hendricks

Answered question

2022-01-04

Derivative of a multivariable function
evaluate the derivative of a function RR defined as
g(t)=f(x+t(yx))
where f:RnR is a multivariable function and x,yRn. Prove that
g(t)=(yx)Tf(x+t(yx))

Answer & Explanation

ramirezhereva

ramirezhereva

Beginner2022-01-05Added 28 answers

Let x=(x1,x2,,xn)andy=(y1,y2,,yn).
Then: x+t(yx)=[x1+t(y1x1),,xn+t(ynxn)].
So, g(t)=f(x1+t(y1x1),,xn+t(ynxn)).
Define zi(t)=xi+t(yixi).
So, g(t)=f(z1(t),..,zn(t)).
Now, g(t)=fz1dz1dt which equals the desired product you have written.

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