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Fill in the blank:
ddx[1−x2]=\frac{d}{dx}\left[\sqrt{1-x^2}\right]=dxd[1−x2]= __________.
−11+x2\frac{-1}{\sqrt{1+x^2}}1+x2−1
11−x2\frac{1}{\sqrt{1-x^2}}1−x21
−11−x2\frac{-1}{\sqrt{1-x^2}}1−x2−1
ddx[1−x2x2+1]=\frac{d}{dx}\left[\frac{1-x^2}{x^2+1}\right]=dxd[x2+11−x2]= __________.
21+x2\frac{2}{1+x^2}1+x22
ddx[(x3)]=\frac{d}{dx}\left[\left(\frac{x}{3}\right)\right]=dxd[(3x)]= __________.
−39−x2-\frac{3}{9-x^2}−9−x23
−39+x2-\frac{3}{9+x^2}−9+x23
−19−x2-\frac{1}{9-x^2}−9−x21
ddx[(xa)]=\frac{d}{dx}\left[\left(\frac{x}{a}\right)\right]=dxd[(ax)]= __________.
−aa2−x2-\frac{a}{a^2-x^2}−a2−x2a
−aa2+x2-\frac{a}{a^2+x^2}−a2+x2a
−1a2−x2-\frac{1}{a^2-x^2}−a2−x21
ddx[2x1−x2]=\frac{d}{dx}\left[\frac{2x}{1-x^2}\right]=dxd[1−x22x]= __________.
−21+x2-\frac{2}{1+x^2}−1+x22
ddx[sinx1+cosx]=\frac{d}{dx}\left[\frac{\sin x}{1+\cos x}\right]=dxd[1+cosxsinx]= __________.
x1+cosx\frac{x}{1+\cos x}1+cosxx
sinx1+cosx\frac{\sin x}{1+\cos x}1+cosxsinx
11+cosx\frac{1}{1+\cos x}1+cosx1
ddx[x2+1x2−1]=\frac{d}{dx}\left[\frac{x^2+1}{x^2-1}\right]=dxd[x2−1x2+1]= __________.
ddx[(ax)]=\frac{d}{dx}\left[\left(\frac{a}{x}\right)\right]=dxd[(xa)]= __________.
−1xx2−a2-\frac{1}{x\sqrt{x^2-a^2}}−xx2−a21
−1a2−x2-\frac{1}{\sqrt{a^2-x^2}}−a2−x21
1a2−x2\frac{1}{\sqrt{a^2-x^2}}a2−x21
ddx[(5x)]=\frac{d}{dx}\left[\left(\frac{5}{x}\right)\right]=dxd[(x5)]= __________.
−1xx2−25-\frac{1}{x\sqrt{x^2-25}}−xx2−251
−125−x2-\frac{1}{\sqrt{25-x^2}}−25−x21
125−x2\frac{1}{\sqrt{25-x^2}}25−x21
If f(x)=(x7)f\left(x\right)=\left(\frac{x}{7}\right)f(x)=(7x). Then ddx[f(x)]=\frac{d}{dx}\left[f\left(x\right)\right]=dxd[f(x)]= __________.
−249−x2-\frac{2}{\sqrt{49-x^2}}−49−x22
−149−x2-\frac{1}{\sqrt{49-x^2}}−49−x21
149−x2\frac{1}{\sqrt{49-x^2}}49−x21
It is done.