a(x) = x^{2}  a'(x) = 2x  
b(x) = sin x  b'(x) = cos x 
y(x) =  a(x) 
b(x) 
y'(x) =  a'(x)•b(x)  a(x)•b'(x) 
b^{2}(x) 
a(x) = 2x  a'(x) = 2  
b(x) = e^{a(x)}  b'(x) = e^{a(x)} = e^{2x}  
c(x) = Sin b(x)  c'(x) = Cos b(x) = Cos e^{2x} 
δSin^{1}x  =  1 
δx  √(1x^{2}) 
δCos^{1}x  =  1 
δx  √(1x^{2}) 
δTan^{1}x  =  1 
δx  1+x^{2} 
Surface area =  ∫  b  2πy√  (1 +  (  dy  )  2  ) dx 
a  dx 
a(x) = r  a'(x)
= 0 (because r is a constant) 

b(x) = √(1  x^{2}/r^{2})  b'(x) = ? 
g(x) = 1   x^{2}  g'(x)
= 2 
x  
r^{2}  r^{2} 
h(x) = √g(x)  h'(x) =  1  
2√(g(x)) 
= 2  x  •  1  =  x 
r^{2}  2√(1  x^{2}/r^{2})  r^{2}√(1 x^{2}/r^{2}) 
b'(x) =  x 
r^{2}√(1 x^{2}/r^{2}) 
=   x 
r√(1 x^{2}/r^{2}) 
(  dy  )  2 
dx 
x^{2}  =  x^{2}  
r^{2}(1 x^{2}/r^{2})  r^{2}  x^{2}  as long as x ≠ r 
1 +  (  dy  )  2 
dx 
1 +  x^{2}  =  (r^{2} x^{2}) + x^{2}  =  r^{2}  
r^{2} x^{2}  r^{2}  x^{2}  r^{2}  x^{2}  as long as r^{2} ≠ x^{2} 
Surface area = 2πr  ∫  r  √(1  x^{2}/r^{2})√  (  r^{2}  )  dx 
r  r^{2}  x^{2} 
Surface area = 2πr  ∫  r  √((  1  x^{2}  )(  r^{2}  ))  dx 
r  r^{2}  r^{2}  x^{2} 
r^{2}  x^{2}  
r^{2}  x^{2}  which equals 1 if r^{2} ≠ x^{2} and √1 = 1 
Surface area = 2πr  ∫  r  1 dx = 2πr  [  x  ]  r 
r  r 
[ln(f(x))]' =  f'(x) 
f(x) 
[ln(f(x))]' =  1 x 
+  1 x 
f'(x) f(x) 
=  1 x 
+  1 x 
f(x) 
=  x^{2}
sin x cos 2x 
now I take the logarithm of this and split it up 
f'(x) f(x) 
=  2x
x^{2} 
+  cos x sin x 
  2 sin 2x cos 2x 
which simplifies to: 
f'(x) f(x) 
=  2 x 
+  cos x sin x 
+ 2 tan 2x  so the next step is to multiply each side by f(x): 
f'(x) =  (  x^{2}
sin x cos 2x 
)(  2 x 
+ cot x + 2 tan 2x  ) 
Have you found an error or do you want to add more
information to these pages? You can contact me at the bottom of the home page. 