MATH100 Lecture Notes - Implicit Function
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MATH100 Full Course Notes
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The chain rule (text 3. 6) y=( f g) ( x)=f [g ( x)] y"=f "( g) . g" ( x) , if u=g( x) dy dx. =(2u) (3 x2)=6 x2( x3+4) y=(x3+4)42(cannot expand 43terms) u=(x3+4) y=u42 y"=(42u41)(3 x2)=126 x2(x3+4)41 x ( 2) y=sin u=x2 y =sin u u cos du dx. = dy du dy dx f ( )=sin 3+sin3 (sin3 =(sin )3) u sin d f " ( )= . We can compose more than 2 functions: (f g h) ( x)=f g[h( x)] (f g h)" (x)= df dg: dg dh, dh dx. Example: (17x ) cos f ( x)= h( x)=17x dh dx. Sin 17 f " ( x)=(5 g4) f ( x)=sin (ex2) f " ( x)=cos (ex 2) ex2 2x. Implicit differentiation (text3. 7) y=f (x) x2+ y2=r2 y= r2 x2(these are 2 functions) y +sin ( x)=x. Thisimplicit function cannot bemade explicit by solving for y. We want beable find derivatives for implicit functionstoo.