# Prove that $f_x(1,1)=-f_y(1,1)$

I have a function $$f(x,y) \in C^1$$. I need to prove that if $$f(x,x)=1$$ for any real number $$x$$, then $$f_x(1,1)=-f_y(1,1)$$.

I tried to prove it using the definition of the partial derivative, but couldn't figure it out.

$$f_x(1,1)=\lim\limits_{h \to 0} \frac{f(1+h,1)-f(1,1)}{h} = \lim\limits_{h \to 0} \frac{f(1+h,1)-1}{h}$$

And the same for $$f_y(1,1)$$.

How can I prove it?

If $$f$$ is differentiable, the directional derivative along $$(1,1)$$ can be computed as $$\partial_{(1,1)} f(1,1) = f'_x(1,1) \cdot 1 + f'_y(1,1)\cdot 1.$$ Since the directional derivative is zero ($$f$$ is constant along that direction), it must be true that $$f'_x(1,1) + f'_y(1,1)=0.$$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 11, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://math.stackexchange.com/questions/3753647/prove-that-f-x1-1-f-y1-1/3753650", "fetch_time": 1621087953000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2021-21/segments/1620243991370.50/warc/CC-MAIN-20210515131024-20210515161024-00344.warc.gz", "warc_record_offset": 422783433, "warc_record_length": 36944, "token_count": 289, "char_count": 714, "score": 3.640625, "int_score": 4, "crawl": "CC-MAIN-2021-21", "snapshot_type": "latest", "language": "en", "language_score": 0.7928414344787598, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00029-of-00064.parquet"}