Using the chain rule to differentiate a multivariate function

Question

Can anyone explain to how to do this derivative. I am given the function

$$f(x, g(x)) = k$$

I have been told by my teacher that the derivative w.r.t $x$ is:

$$f_x + g'(x)f_y$$

He didn't explain it very well so it would be very helpful if someone could explain this to me in simple terms.

Answer 1

This is a consequence of:

• On the one hand the chain rule as the derivative of the composition of functions $t \mapsto f(x, g(t))$ is $g^\prime(t) f_y(x,g(t))$.
• On the other hand the (Fréchet) derivative of the map $f(x,y)$ is given by $f^\prime(x,y)(h,k)=f_x(x,y)h+ f_y(x,y)k$.