How to Differentiate a Complex Function: Tips and Tricks for Success

[Millacol88]

Does somebody want to help me with differentiating this function:

(0.003 + 0.004\sqrt{2})∏r² + 0.004∏r [(2250 - ∏r³) / 3∏r²)]

Originally I tried simplifying it by distributing the 0.004∏r into the fraction, but whenever I did that the two expressions were never equivalent when I checked them on a graphing calculator. Then I tried using the product and quotient rules to differentiate the second term and the power rule for the first, but based on the graph that was not the correct derivative. I found that the derivative had only one zero at -1 which doesn't make sense because the original function clearly has a local minimum at around x = 4. Any help is appreciated, this is really bugging me. :s

 

[SammyS]

Does somebody want to help me with differentiating this function:

(0.003 + 0.004\sqrt{2})∏r² + 0.004∏r [(2250 - ∏r³) / 3∏r²)]

Originally I tried simplifying it by distributing the 0.004∏r into the fraction, but whenever I did that the two expressions were never equivalent when I checked them on a graphing calculator. Then I tried using the product and quotient rules to differentiate the second term and the power rule for the first, but based on the graph that was not the correct derivative. I found that the derivative had only one zero at -1 which doesn't make sense because the original function clearly has a local minimum at around x = 4. Any help is appreciated, this is really bugging me. :s

What did you get for your function when you did the following?

Originally I tried simplifying it by distributing the 0.004∏r into the fraction, but ...

 

[Reptillian]

Are you sure there wasn't an algebra mistake? Be sure to double check your signs, a missed negative sign can be disasterous. Also double check your powers.

Did you notice that after the plus sign, the pi and r cancel with the pi and one r in the denominator? Makes it so you only need to use the quotient rule in the second term.

Before differentiation, it simplifies to: N*r^2 - M*r +L*(1/r) where N, M, and L are just numbers.

 

[Millacol88]

After simplifying I get:
(0.003 + 0.004√2) ∏r² + 3/r - (4∏r²) / 300

Which is equivalent now according to a graphing calculator.

Differentiating, I get:
(0.006 + 0.008√2) ∏r - 3r^-2 - 1500/∏r³

But that's not right. i.e. it has positive values when the slope of the function is negative. :/

 

[Reptillian]

Sorry, I missed an r in my earlier calculation...

 

[Millacol88]

Yeah I noticed. No worries. Do you know what's wrong with the derivative. I found just with the power rule and the quotient rule on the last term.

 

[Reptillian]

Where did you get that 4/300 from? Shouldn't it be 1/750?

The 4/300 has a decimal expansion of 0.01333333... and the 1/750 is 0.0013333... I think you missed a zero. :)

 

[Millacol88]

I typed that wrong, I did get 4/3000 (1/750).

 

[Reptillian]

Ok...that makes sense. :) Where is that 1/r^3 coming from in your derivative?

 

[Millacol88]

Using the quotient rule on the 4∏r² / 3000 term. The denominator of the differentiated expression is 16∏²r⁴. A pi and an r cancel from the numerator which was 8∏r(3000).

 

[Reptillian]

Quotient rule? You have the first two terms in the derivative right...the r and r^-2 terms...but the last one -1500/(pi*r^3) is wrong. That term should be your -8*pi*r/3000. You shouldn't need the quotient rule at all, just the power rule.

 

[SammyS]

Does somebody want to help me with differentiating this function:

(0.003 + 0.004\sqrt{2})∏r² + 0.004∏r [(2250 - ∏r³) / 3∏r²)]

Originally I tried simplifying it by distributing the 0.004∏r into the fraction, but whenever I did that the two expressions were never equivalent when I checked them on a graphing calculator. Then I tried using the product and quotient rules to differentiate the second term and the power rule for the first, but based on the graph that was not the correct derivative. I found that the derivative had only one zero at -1 which doesn't make sense because the original function clearly has a local minimum at around x = 4. Any help is appreciated, this is really bugging me. :s

After simplifying I get:
(0.003 + 0.004√2) ∏r² + 3/r - (4∏r²) / 300

Which is equivalent now according to a graphing calculator.

Differentiating, I get:
(0.006 + 0.008√2) ∏r - 3r^-2 - 1500/∏r³

But that's not right. i.e. it has positive values when the slope of the function is negative. :/

It looks like you forgot to multiply by 0.004 .

 

[Millacol88]

It looks like you forgot to multiply by 0.004 .

No, I multiplied that in at the very beginning. I scaled up the fraction by 1000 times so I had a 4 in the numerator rather than a decimal. After fixing what Reptilian mentioned, my derivative looks correct, and the zero is at the x-value of that local minimum, so I think it's right.