關於微分的證明

Proof of the power rule
To prove the power rule for differentiation, we use the definition of the derivative as a limit:


圖片參考：http://upload.wikimedia.org/math/b/4/2/b427bcd6466525e0b402f7ee72bb1a91.png

Substituting f(x) = xn gives


圖片參考：http://upload.wikimedia.org/math/7/0/4/7043c96f85a837f89205205e30eb8a10.png

One can then express (x + h)n by applying the binomial theorem to obtain


圖片參考：http://upload.wikimedia.org/math/b/3/4/b34fea30575f7631a2c7a11db3e9b08e.png

The i = n term of the sum can then be written independently of the sum to yield


圖片參考：http://upload.wikimedia.org/math/b/8/3/b834303e09297692808e802c6177ca57.png

Canceling the xn terms one generates


圖片參考：http://upload.wikimedia.org/math/7/c/7/7c74dabaa4491cddb90ba1a8713f5df3.png

An h can be factored out from each term in the sum to give


圖片參考：http://upload.wikimedia.org/math/d/6/d/d6d233f83bd3bc879fe59a0a845f4ab7.png

From thence we can cancel the h in the denominator to obtain


圖片參考：http://upload.wikimedia.org/math/7/8/f/78f6d6da126b992f0b79232c6d2301b4.png

To evaluate this limit we observe that n − i − 1 > 0 for all i < n − 1 and equal to zero for i = n − 1. Thus only the h0 term will survive with i = n − 1 yielding


圖片參考：http://upload.wikimedia.org/math/9/a/2/9a210085037325bbecdc9d281e942d39.png

Evaluating the binomial coefficient gives


圖片參考：http://upload.wikimedia.org/math/6/4/1/6419acd6b28e944904721da56864880f.png

It follows that


圖片參考：http://upload.wikimedia.org/math/f/9/0/f901175a9fbabecc263448043a01f03a.png

Proof of the product rule
A rigorous proof of the product rule can be given using the properties of limits and the definition of the derivative as a limit of Newton's difference quotient:
Suppose


圖片參考：http://upload.wikimedia.org/math/a/2/3/a2374d480e0d349e52d9bee469294cce.png

and suppose further that g and h are each differentiable at the fixed number x. Then






圖片參考：http://upload.wikimedia.org/math/1/f/b/1fb2ed0abb69cff24248abd7a8ab1faf.png


圖片參考：http://upload.wikimedia.org/math/4/0/5/4054f43fa54b0b1881fe6d2a111c51fd.png




圖片參考：http://upload.wikimedia.org/math/8/a/8/8a84f149e1c7a705fee04ec38d1df775.png




圖片參考：http://upload.wikimedia.org/math/7/e/e/7eee95801b35733361158c5ea2484549.png




圖片參考：http://upload.wikimedia.org/math/a/2/4/a24339bbcac1e21efb9703fb5b361f5e.png




圖片參考：http://upload.wikimedia.org/math/5/6/8/5687225f3e491bf7a9260cad650eafda.png

Since h is continuous at x, we have


圖片參考：http://upload.wikimedia.org/math/2/c/e/2ceabc078f7efa4d5bff55c697fc36b3.png

and by the definition of the derivative, and the differentiability of g and h at x, we also have


圖片參考：http://upload.wikimedia.org/math/2/f/3/2f38890d9d80bfce24f9871117714dc7.png


圖片參考：http://upload.wikimedia.org/math/b/4/a/b4a9264b7a614b41eee52c0074ec0cda.png

Thus, we are justified in splitting each of the products inside the limit, and putting everything together, and we have


圖片參考：http://upload.wikimedia.org/math/5/0/4/50404304b95afeb3da1b6208209a0fbf.png



圖片參考：http://upload.wikimedia.org/math/0/b/c/0bcf0c385986002772053bbb230a3dbe.png



圖片參考：http://upload.wikimedia.org/math/6/c/2/6c24a0bac69540f4d3ac90d7efa3a0c4.png

and this completes the proof.
Proofs

From Newton's difference quotient

Suppose f(x) = g(x) / h(x)

where h(x)≠ 0 and g and h are differentiable.


圖片參考：http://upload.wikimedia.org/math/e/a/a/eaa454a58cb7f7f55bf8722914eb9225.png



圖片參考：http://upload.wikimedia.org/math/c/c/3/cc30d166c0d53a6824f068b76159a1b4.png



圖片參考：http://upload.wikimedia.org/math/2/8/2/282b0126b22bd619b53a6a2ad0efbe57.png



圖片參考：http://upload.wikimedia.org/math/e/d/6/ed62e7ecfdb50f5908bf2496ae9e5da1.png



圖片參考：http://upload.wikimedia.org/math/7/9/0/790fad4d62605c588832cc670060032d.png



圖片參考：http://upload.wikimedia.org/math/e/8/6/e868194a474129b8baca7ca0905642c7.png



圖片參考：http://upload.wikimedia.org/math/7/9/4/79455cf2bc5ba292947ea4e5fa92ab3f.png


From the product rule

Suppose
圖片參考：http://upload.wikimedia.org/math/8/5/c/85c4e200964ebf85acaee18f2c1f03f3.png



圖片參考：http://upload.wikimedia.org/math/c/3/e/c3e9ad796dbb5c4b9cc7c2ba783035b4.png



圖片參考：http://upload.wikimedia.org/math/f/e/a/fea4655ead4645f4b89d862304116463.png

The rest is simple algebra to make f'(x) the only term on the left hand side of the equation and to remove f(x) from the right side of the equation.


圖片參考：http://upload.wikimedia.org/math/e/6/5/e65528ef4586fbfa2d4461ac6127150e.png



圖片參考：http://upload.wikimedia.org/math/a/f/6/af63cc5df5f1e90096c1df20e47a461e.png

Proof of the chain rule
Let f and g be functions and let x be a number such that f is differentiable at g(x) and g is differentiable at x. Then by the definition of differentiability,


圖片參考：http://upload.wikimedia.org/math/0/4/2/0429b73452218c9c59ec40cc8b0aed28.png

Similarly,


圖片參考：http://upload.wikimedia.org/math/9/5/9/9591538a0da0b710bbfce9c8af1849d2.png

Now






圖片參考：http://upload.wikimedia.org/math/6/9/e/69e8004e558c50e71748892d2b618b66.png


圖片參考：http://upload.wikimedia.org/math/7/0/4/704a9a7e51c33f8d1aaba4eaa5c56d4a.png




圖片參考：http://upload.wikimedia.org/math/d/c/3/dc357e723b1571c648785a871acf0533.png

where
圖片參考：http://upload.wikimedia.org/math/9/8/3/983747a0162b1c6896675999239dcca7.png
. Therefore


圖片參考：http://upload.wikimedia.org/math/f/b/0/fb098445b3691488ce1133e126197da2.png