Differentiating x(t): Solutions & Examples

[JakePearson]

1)
Differentiate x(t) = pi / t²
pi x(t) = t^-2
= pi (t^-2)
= -2pi t^-3

2)
Differentiate x(t) = 1 / (At³ + B)
= (At³ + B)^-2 x (2At²)
= (-2At²) / (At³ + B)

3)
Differentiate x(t) = [A sqrt(t+B)]⁴
= [A (t+B)^1/2]⁴
= A⁴(t + B)²
= 2A⁴(t + B)

4)
The nebraska board of grain are designing new portable grain silos. they have enough sheet material to make 2000 cylinderical containers, each of fixed surface area 54m² (this includesthe cylinder ends). calculate in terms of pi the maximum volume of grain that could be stored in total?

my answer is = for 1 container V is 30.46m³, so for 2000 containers (30.46 x 2000) = 60920m³ is this correct

5)
The height h(x) in meters above the ground of a parachute varies with her horizontal distance x in meters from a landing target on the ground as h(x) = 50sin^-1 (0.1x). What is the rate of change of h with respect to x = 6m?

my answer is (25 / 4) is this correct

 

[tiny-tim]

HI JakePearson!

First, you must begin your proofs with "dx/dt =" (or "x'(t) = ").

(and have a pi: π and a square-root: √ )

1)
Differentiate x(t) = pi / t²
pi x(t) = t^-2
= pi (t^-2)
= -2pi t^-3

ok (apart from "pi x(t) = t^-2")

2)
Differentiate x(t) = 1 / (At³ + B)
= (At³ + B)^-2 x (2At²)
= (-2At²) / (At³ + B)

No, 2At² is wrong, and the last line is also wrong.

3)
Differentiate x(t) = [A sqrt(t+B)]⁴
= [A (t+B)^1/2]⁴
= A⁴(t + B)²
= 2A⁴(t + B)

ok, except see my original comment.

4) …

5) …

uhh? show your calclulations!

 

[g_edgar]

1) is OK

2) is wrong ... I assume there is a - sign error of copying, since it appears later. But there is another error.

3) is OK

4) is at least incomplete. To tell if if it right I would have to do it myself (rather than checking that you have done it right.

5) correct answer, but if "showing work" is required it is incomplete

 

[Mark44]

1)
Differentiate x(t) = pi / t²
pi x(t) = t^-2
= pi (t^-2)
= -2pi t^-3

You arrived at the correct answer, but the work you show is incorrect. To expand on what tiny-tim said, your first line should be the function you're going to differentiate, and the second should start with dx/dt or x'(t).

Your second line is incorrect. Apparently you multiplied both sides by $\pi$ (which would leave $\pi^2$ on the right side. The expression in the third line is not equal to the previous line. What you omitted showing is that you took the derivative.

Here's how your work should look:
x(t) = $\pi$ / t² = $\pi$ t^-2
x'(t) = -2 $\pi$ t^-3

It's very important to distinguish between the equation for the function you're differentiating and the equation for the derivative. If you mix them up into one big, amorphous glop, there will come a time -- I GUARANTEE IT--that it will come along and bite you in the butt.