Explain why does gradient vector points in the direction of the steepest increase?

Gradient vector is build of partial derivatives of the function of multiple variables. But what is the intuition behind the statement that "gradient vector points in the direction of the steepest increase"? If I have a function $$f(x,y)$$
where $$f_x(x,y)=\frac{df}{dx}$$ where we just treat y as a constant, and differentiate the function with respect to x. It gives me the rate of change along x axis. The same for y. But what I don't understand is why does a vector combined of those two partial derivatives points at the direction of the steepest increase? I want to receive a comprehensive answer, possibly with graphs.

  • I need it in 12 hours please. Timer was reset and I didn't notice it.

    • I edited the deadline to 12 hours.

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Kav10 Kav10
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  • @ Kav10: I edited your answer, and used LaTeX for typesetting the formulas. Take a look at the revised source and see the changes I have made. Please use LaTeX for typesetting formulas in your answers to make them more readable. Thank you very much for your carefully written solutions and your contributions to the Matchmaticians community.

    • Kav10 Kav10
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      Thank you, Walter! I appreciate it. I am still not used to typesetting the formulas with LaTeX and hence I either attach a document with formulas written in Word or when have less time, just write them here in the response box. I'll try to use LaTex more for typesetting formulas. Thanks for the support. A couple of months ago, I sent a long list of ideas to improve things on the website and make things easier for the users. I am hoping you guys are working on at least some of them if not all.

  • Thanks. You can click "View Raw" in your accepted questions and see the source file. You'll learn typesetting with LaTeX very quickly.

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