Derivative of a multivariable function evaluate the derivative of a function R→R defined as g(t)=f(x+t(y−x)) where f:Rn→R is a multivariable function and x,y∈Rn. Prove that g′(t)=(y−x)T∇f(x+t(y−x))

Question

Derivative of a multivariable function  evaluate the derivative of a function R→R defined as  g(t)=f(x+t(y−x))  where f:Rn→R is a multivariable function and x,y∈Rn. Prove that  g′(t)=(y−x)T∇f(x+t(y−x))

ramirezhereva · Accepted Answer

Let x=(x1,x2,…,xn)andy=(y1,y2,…,yn).Then: x+t(y−x)=[x1+t(y1−x1),…,xn+t(yn−xn)].So, g(t)=f(x1+t(y1−x1),…,xn+t(yn−xn)).Define zi(t)=xi+t(yi−xi).So, g(t)=f(z1(t),..,zn(t)).Now, g′(t)=∑∂f∂z1dz1dt which equals the desired product you have written.

Derivative of a multivariable function evaluate the derivative of a function R→R defined as g(t)=f(x+t(y−x))...

Answered question

Answer & Explanation