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Can the derivative rules be proven geometrically?

Consider the following graph:
enter image description here
y1 = x2

y2 = 4x2
the derivative of y1 and y2 is: $2x$ and $8x$ respectively.
If z = y1 + y2, then dy/dx = d(y1)/dx + d(y2)/dx;
This is actually the function 10x,

there are analytical proofs of the sum rule, product rule, quotient rule of derivatives. But, is there a geometrical proof of these rules? Particularly of the sum, quotient and the product rules of derivatives.

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    The derivative of y2 isn't 10x... it's 4*2*x = 8x – 2017-02-19
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    @JuanIgnacioCuiule, yes it's 8x but that added to 2x is 10x – 2017-02-20
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    @JuanIgnacioCuiule, I had mentioned it above. ;-) – 2017-02-20
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    "the derivative of y1 and y2 is: 2x and 10x respectively." you wrote this... – 2017-02-20
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    @JuanIgnacioCuiule, thanks for the edit, didn't spot the typo. – 2017-02-20

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