# Multivariate function to univariate function

I'm currently trying to understand the proof of the directional derivative of a multivariate function $$f(x,y)$$ at the point $$(x_0,y_0)$$ along the vector $$\vec u \langle a,b \rangle$$.

$$D_\vec u f(x_0,y_0) = \lim_{h\to0} \dfrac{f(x_0 + ah, y_0 + bh) - f(x_0, y_0)}{h}$$

Then, it says let's put:

$$g(h) = f(x_0 + ah, y_0 + bh)$$

How can we change a function of 2 variables to a function of only one variable, which is a totally different one. Is it really valid? How is this process called, I couldn't find the exact name to search for more details. Thank you.

## 1 Answer

Pick a function, $$f(x,y)=x^2+y^2$$, a point, $$P(x_0,y_0)=(1,2)$$ and a direction $$\vec u=(3,4)$$, then calculate the derivative of $$f$$ at $$P$$ along $$\vec u$$

$$D_\vec u f(x_0,y_0) = \lim_{h\to0} \dfrac{f(x_0 + ah, y_0 + bh) - f(x_0, y_0)}{h}$$

$$D_{(3,4)} f(1,2) = \lim_{h\to0} \dfrac{(1 + 3h)^2+(2 + 4h)^2 - (1^2+2^2)}{h}$$

Clearly $$f(x_0 + ah, y_0 + bh)=(1 + 3h)^2+(2 + 4h)^2$$ is a one variable function in $$h$$. We can call it $$g$$, so is

$$g(h)=(1 + 3h)^2+(2 + 4h)^2=25h^2+22h+5$$

$$g$$ is the value of $$f$$ fixing a point and moving along a fixed direction. The procedure is part, as you see, of the construction of the incremental quotient, the very basis of the differential calculus and, to me, a useful starting point for the intuition about the subject.

• It is much more clear said this way. Thank you! – Ryan B. Nov 23 '18 at 16:30
• You are welcome :) – Rafa Budría Nov 23 '18 at 16:31