Expert: Paul Klarreich - 12/8/2006

Question
1). Calculate dy/dx if y=4x³+3x²+2

A.12x² + 6x
B.12x²+6x+2
C.12x³+6x²
D.12x³+6x²+2

2).What is the derivative of y with respect to x if y=2/x^4

A.-8/x^5
B.-8/x^3
C.8/x^3
D.8/x^5

3).What is the derivative of y =√3x

A.dy/dx=√3x/6x
B.dy/x=√3x/2x
C.dy/dx=2√3x/3x
D.dy/dx=√3x/2

4).Calculate the derivative of y=√7x+2

A.1/√7x
B.1/2√7x+2
C.7/2√7x+2
D.√7/√7x+2

5).If y=x³+3x²+5, what is dy/dx

A.3x²+6x
B.3x²+6x+5
C.6x²
D.6x²+5

6).If a body moves in a straight line according to the law s=24t+3t²-t³, where s is the distance measured in meters from the orgin and t is the time in seconds after it starts to move,calculate the body's velocity as a function of time.

A.v=30t-3t²
B.v=30t-t²
C.v=24+3t-t²
D.v=24+6t-3t²

7).What is the velocity of the body in question 6 when t=3 seconds?

A.15 m/s
B.27 m/s
C.63 m/s
D.81 m/s

Answer
Questioner:  Sonny
Category:  Calculus
 
Question:  
1). Calculate dy/dx if y=4x³+3x²+2

A.12x² + 6x
B.12x²+6x+2
C.12x³+6x²
D.12x³+6x²+2

2).What is the derivative of y with respect to x if y=2/x^4

A.-8/x^5
B.-8/x^3
C.8/x^3
D.8/x^5

3).What is the derivative of y =sqrt(3x

A.dy/dx=sqrt(3x/6x
B.dy/x=sqrt(3x/2x
C.dy/dx=2sqrt(3x/3x
D.dy/dx=sqrt(3x/2

4) Calculate the derivative of y=sqrt(7x)+2

A.1/sqrt(7)x
B.1/2sqrt(7)x+2
C.7/2sqrt(7)x+2
D.sqrt(7)/sqrt(7)x+2

5).If y=x³+3x²+5, what is dy/dx

A.3x²+6x
B.3x²+6x+5
C.6x²
D.6x²+5

6).If a body moves in a straight line according to the law s=24t+3t²-t³, where s is the distance measured in meters from the orgin and t is the time in seconds after it starts to move,calculate the body's velocity as a function of time.

A.v=30t-3t²
B.v=30t-t²
C.v=24+3t-t²
D.v=24+6t-3t²

7).What is the velocity of the body in question 6 when t=3 seconds?

A.15 m/s
B.27 m/s
C.63 m/s
D.81 m/s
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Hi, Sonny,

Before I start, I have to tell you that the interface at this site is clumsy and I have no good way to make certain of the symbols.  So please observe certain conventions:

1. Write  x^n  for  "x to the n", as in x^3 for "x-cubed".

2. Always, always, always parenthesize fully so your expression is clear.  Don't write 1/2x, for example, because it is not clear whether you mean:
1      1
--- or --- x
2x      2

I'm going to have to make certain assumptions about your examples, since you didn't know this before.  MAKE SURE THE PROBLEM I'M DOING IS THE SAME AS YOURS.

And one more thing -- try to make the subject line refer to the question that you are asking.  "Calculus Help" says nothing at all.  Of course this is calculus; that's the name of the section, and of course you need help, otherwise you wouldn't be asking.  [In fact, this might help you understand the examples better, too.]


1). Calculate dy/dx if y=4x^3+3x^2+2

This is just a polynomial and should be routine:

dy/dx = 4(3x^2) + 3(2x) = 12x^2 + 6x

2) What is the derivative of y with respect to x if y=2/(x^4)

y = 2x^(-4)

dy/dx = 2(-4)x^(-3) = -8x^(-3)
  -8
= -----
  x^3


3).What is the derivative of y = sqrt(3x)

y = (3x)^1/2

dy/dx = (1/2)(3x)^-1/2(3)   <<< 3 from the chain rule.
     3
= ----------
 2 sqrt(3x)

4).Calculate the derivative of

A. Is it  y=sqrt(7)x+2 ??
B. Is it  y=sqrt(7x)+2 ??
C. Is it  y=sqrt(7x+2) ??

These are all different, and if you don't parenthesize, then it is not clear.

I will assume it is C:

y=sqrt(7x+2) = (7x + 2)^1/2

dy/dx = (1/2)(7x + 2)^-1/2(7)   << 7 from the chain rule.
        7
= ---------------
 2 sqrt(7x + 2)

5).If y=x^3 + 3x^2 + 5, what is dy/dx

Another polynomial, which should be routine.

dy/dx = 3x^2 + 3(2x) = 3x^2 + 6x

6).If a body moves in a straight line according to the law s=24t+3t²-t³, where s is the distance measured in meters from the orgin and t is the time in seconds after it starts to move,calculate the body's velocity as a function of time.

If  s(t) = 24t  + 3t^2 - t^3,  

v(t) = ds/dt = 24 + 6t - 3t^2

7) What is the velocity of the body in question 6 when t=3 seconds?

v(3) = 24 + 6(3) - 3(3)^2
v(3) = 24 + 18 - 3(9)
v(3) = 42 - 27 = 15