# Thread: Differentiate ε = 4x10^−5t^4−0.0024t^3+0.0487t^2−0.3428t+1.4532

1. ## Differentiate ε = 4x10^−5t^4−0.0024t^3+0.0487t^2−0.3428t+1.4532

Hi guys,

The following equation was derived from a theoretical test, where t is equal to time:

ε = 4x10^−5t^4−0.0024t^3+0.0487t^2−0.3428t+1.4532

I was told I need to differentiate to dε/dt

I have got the following as my answer:

(4t^3−180t^2+2435t−8570) / 25000

Does that look correct?

I was then asked what the answer would be if the time was 15 hours, which I would then put in to the above equation - if it is correct?

Thanks for any pointer.

2. Well, I'm assuming the superscript exponents didn't carry over from when you copied-and-pasted, and you actually meant:

$\epsilon = 4 \cdot 10 - 5t^4 - 0.0024t^3 + 0.0487t^2 - 0.3428t + 1.4532$

$\dfrac{d\epsilon}{dt} = \dfrac{4t^3 - 180t^2 + 2435t - 8570}{25000}$

If that's the case, your derivation is almost correct, except for the 4t3 term. Recall the power rule tells us that $\dfrac{d}{dt} \left( t^\alpha \right) = \alpha \cdot t^{\alpha - 1}$ and the constant multiple rule tells us that $\dfrac{d}{dt} \left( C \cdot f(t) \right) = C \cdot \dfrac{d}{dt} \left( f(t) \right)$. Combining these two rules, what does that make $\dfrac{d}{dt} \left( 5t^4 \right)$? (Hint: It's not 4t3/25000 )

As for your second query, assuming that the declaration of the variable was something like "t is the time, measured in hours, since...", then yes, you're trying to find the derivative at the point t = 15. Since the derivative of a function is itself a function, you can simply plug in the desired value and evaluate.

3. Hi, Thank you for the reply. Yes, I don't think the equation copied properly. The original equation was:

. . .$\epsilon\, =\, 4\cdot 10^{-5}\,t^4\, -\, 0.0024\, t^3\, +\, 0.0487\, t^2\, -\, 0.3428\, t\, +\, 1.4532$

My derivative was:

. . .$\dfrac{d\epsilon}{dt}\, =\, \dfrac{4t^3\, -\, 180t^2\, +\, 2435t\, -\, 8570}{25000}$

Hopefully this shows OK for what I was trying to explain?

4. Ah, okay. Armed with this correction, your derivative is absolutely correct. What did you get for the next step, when you plug in t = 15?

5. Originally Posted by ksdhart2
Ah, okay. Armed with this correction, your derivative is absolutely correct. What did you get for the next step, when you plug in t = 15?
Thanks for that

I got 0.0382 as my answer.

6. That "/25000" form is a bit misleading. It gives an aura of exactness that doesn't actually exist. If you start with four decimal places, you probably should stay with the decimal places, rather than converting to a cleaner-looking picture.

7. Originally Posted by tkhunny
That "/25000" form is a bit misleading. It gives an aura of exactness that doesn't actually exist. If you start with four decimal places, you probably should stay with the decimal places, rather than converting to a cleaner-looking picture.
Hi tkhunny,

If I didn't include the /25000, what would be the best way to present the equation? Apologies, I don't understand.

Thank you
Ryan

8. Originally Posted by mcarthyryan
Hi guys,

The following equation was derived from a theoretical test, where t is equal to time (in what unit? seconds? minutes? hours? days?.:

ε = 4x10^−5t^4−0.0024t^3+0.0487t^2−0.3428t+1.4532

I was told I need to differentiate to dε/dt

I have got the following as my answer:

(4t^3−180t^2+2435t−8570) / 25000

Does that look correct?

I was then asked what the answer would be if the time was 15 hours, which I would then put in to the above equation - if it is correct? (answer to this question will depend on answer to that question)

Thanks for any pointer.
.

9. Originally Posted by Subhotosh Khan
.
Hi,

in hours.

10. Originally Posted by mcarthyryan
If I didn't include the /25000, what would be the best way to present the equation?
I'm not sure there's an issue, with your form.

If we change all decimal numbers to their Rational form, and we combine everything into a single ratio, then 25000 is the common denominator.

Perhaps, when rounding your final result (when t=15), you don't have enough precision from the givens to justify reporting to four decimal places (see Significant Figures). You'll need to inquire, whether that's a concern.

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