Calculus I Assignment # 2 
This assignment covers
Derivatives with power, product, quotient and chain rule
Implicit Differentiation
Higer Order Derivatives
QUESTIONS
A) Find the first derivative
1)  2)  3) 
4)  5) f (x) = sin^{ 2} (4x^{ 3})  6) 
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B) Use implicit differentiation to find the derivative
1) 5x^{ 3}  2x^{ 2} y^{ 2} + 4y^{ 3}  7 = 0  2) 3x^{ 2}  x y^{ 2} + y^{  1} = 1  3) xy^{ 2} = sin (x + 2y) 
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C)
1) Find equations of the tangent and normal lines to x^{ 2} y  y^{ 3} = 8 at P(3, 1)
2) Find the first, second and third derivative of y = 5x^{ 3} + 4x^{ 1 / 2}
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SOLUTIONS
A) Find the first derivative
. 1) 
2) 
3) 
4) 
5) f '(x) = 24x^{ 2} sin (4x^{ 3})cos (4x^{ 3}) 
6) 
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B) Use implicit differentiation to find the derivative
1) 15x^{ 2}  4xy^{ 2}  4x^{ 2} yy' + 12y^{ 2} y' = 0 factor out y', transpose and divide to get

2) to simplify, multiply through by y^{ 2}.

3) 
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C)
1) 2xy + x^{ 2} y' 3y^{ 2} y' = 0
we solve for y' and substitute x = 3, y = 1
The equation of the normal line with slope = 1 is y = x 2.
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2)
y' = 15x^{ 2} + 2x^{  1 / 2}  y'' = 30x^{ }  x^{  3 / 2}  y''' = 30 + (3/2) x^{  5 / 2} 
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