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Let $f : \mathbb R^2 \longrightarrow \mathbb R$ be function of two real variables $x$ and $y$ and $x$ and $y$ are functions of a single variable $t$.Let $x$ and $y$ are differentiable at $t = a$ and let $f$ is differentiable at $(x(a),y(a))$.Then

$$\frac {df} {dt} \vert_{(x(a),y(a))} = \frac {\partial f} {\partial x} \vert_{(x(a),y(a))} . {\frac {dx} {dt}} {\vert_{t = a}} + {\frac {\partial f} {\partial y}} {\vert_{(x(a),y(a))}} . {\frac {dy} {dt}} {\vert_{t = a}}$$.

My attempt :

$${\frac {df} {dt} \vert_{(x(a),y(a))}} = {\lim_{t \rightarrow 0}} {\frac {f(x(a + t),y(a + t)) - f(x(a),y(a))} {t}} = {\lim_{t \rightarrow 0}} {\frac {f(x(a + t),y(a + t) - f(x(a),y(a + t)) + f(x(a),y(a + t)) - f(x(a),y(a))} {t}} = {\lim_{t \rightarrow 0}} {\frac {f(x(a + t),y(a + t)) - f(x(a),y(a + t))} {x(a + t) - x(a)}} . {\frac {x(a + t) - x(a)} {t}} + {\lim_{t \rightarrow 0}} {\frac {f(x(a),y(a + t)) - f(x(a),y(a))} {y(a + t) - y(a)}} . {\frac {y(a + t) - y(a)} {t}}$$.

Now I have stuck.I don't know what is the limit

$$\lim_{t \rightarrow 0} {\frac {f(x(a + t),y(a + t)) - f(x(a),y(a + t))} {x(a + t) - x(a)}}$$

Is it $$\frac {\partial f} {\partial x} \vert_{(x(a),y(a + t))}$$?Please help me.

I really find difficulty here.

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    Is continuity of $x$ and $y$ important? – 2017-01-01
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    What will happen if $x$ is a constant function? – 2017-01-01
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    Your attempt looks fine to me. Now you just need to use -- as you commented correctly -- that $x$ and $y$ are continuous at $a$ (because they are differentiable at $a$). This will imply the desired equality. Do you know how to prove this? – 2017-01-01
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    No. I can't proceed further @m. s. because of the limit which was mentioned at the end of my post.My question is it equal to $${\frac {\partial f} {\partial x}} {\vert_{(x(a),y(a + t))}}$$? Please explain. – 2017-01-01

1 Answers 1

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In the undergraduate sophomore courses I have seen, the textbooks would frequently approximate your limit as $\frac {df} {dx}$ at $(x (a), y (a))$.

A more thorough, formal treatment incorporates changes to definition of limits.

One proper way to define a derivative or limit from $\mathbb R \times \mathbb R$ to ${\mathbb R}$ starts with a circular domain with center $( x (a), y(a))$ and radius = $max. ( x (a+t)−x(a), y(a+t)−y(a))$.

Another proper way to define a derivative or limit from $\mathbb R \times \mathbb R$ to $\mathbb R$ starts with a domain of $[x (a-t), x(a+t)] \times [y (a-t), y(a+t)]$ depending on your how picky your professor is, what level of course you are in, etc.

For simplicity of notation, I'm assuming $x (a-t) < x (a+t)$, and similar for $y$.

A full, authoritative answer may require Riemann Stieltjes sums, maximum of infinimums, and more. At my alma mater, these concepts are taught in graduate analysis.