implicit derviv w/ natural log

calcstruggles2013

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Mar 11, 2013
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6
y =
e3x^2 (8 x5+14)4

(21 x2+5 x−10)2

"Find the derivative."
Lon capa suggests to take the natural log of both sides but I don’t understand how that would help? Bc then you would have
3x^2(8x^5+14)^4 divided by (21x^2 +5x-10)^2
Or would multiplying by the natural log give you
3x^2*4(8x^5+14) divided by 2(21x^2+5x-10)
Also, on the other side wouldn’t you have y’/y so at the end you would have to multiply the right side by the original equation to get y’ alone?
I went about it the second way because I thought it seemed easier and I got an insanely long answer
(((84x+10(6x(32x^5+56))-(42x^2+10x-20)(6(32x&5+56)+6x(160x^4))+((84x+10(160x^4(3x^2))-(640x^3(3x^2)+(6x(160x^4)))(42x^2+10x-20)))*(exp(3x^2)(8x^5+14^4)/(21x^2+5x-10)^2)
I also had my friend try and he got:
(((exp(3*(x^2)))*(8*(x^5)+14)^4)/((21*(x^2)+5x-10)^2))*(6*x+((160*(x^4)/(8*(x^5)+14))+(84*x+10)/(21*(x^2)+5*x-10)))
 
y =

e3x^2 (8 x5+14)4

(21 x2+5 x−10)2

"Find the derivative."
Lon capa suggests to take the natural log of both sides but I don’t understand how that would help? Bc then you would have
3x^2(8x^5+14)^4 divided by (21x^2 +5x-10)^2
Or would multiplying by the natural log give you
3x^2*4(8x^5+14) divided by 2(21x^2+5x-10)
Also, on the other side wouldn’t you have y’/y so at the end you would have to multiply the right side by the original equation to get y’ alone?
I went about it the second way because I thought it seemed easier and I got an insanely long answer
(((84x+10(6x(32x^5+56))-(42x^2+10x-20)(6(32x&5+56)+6x(160x^4))+((84x+10(160x^4(3x^2))-(640x^3(3x^2)+(6x(160x^4)))(42x^2+10x-20)))*(exp(3x^2)(8x^5+14^4)/(21x^2+5x-10)^2)
I also had my friend try and he got:
(((exp(3*(x^2)))*(8*(x^5)+14)^4)/((21*(x^2)+5x-10)^2))*(6*x+((160*(x^4)/(8*(x^5)+14))+(84*x+10)/(21*(x^2)+5*x-10)))

Yes, take the natural log of both sides then apply the log rules that you should have been taught. Do you remember them? Once you apply the rules then taking the derviative of a bunch of nalutral log terms is much more simpler than doing the quotient rule to the behemoth that you were given. Remember that you also have to take the derivative of ln(y) as well.
 
y = e3x^2 (8 x5+14)4

(21 x2+5 x−10)2

"Find the derivative."
Since you are told to use implicit differentiation, that is a clue to make the expression implicit. Right now it is y = f(x), which is explicit. First thing I would do is multiply by that denominator:

\(\displaystyle \displaystyle y\ \left( 21 x^2 + 5x - 10 \right)^2 = e^{3 x^2}\ \left(8x^5 + 14\right)^4\)

Now it really is implicit. Next step should be to take logarithms, which will make 2 exponents constants, as well as bring 3x^2 down to earth.

Let us know how far you get. If you get stuck, we may be able to offer more hints.
 
y =

e3x^2 (8 x5+14)4

(21 x2+5 x−10)2

"Find the derivative."
Lon capa suggests to take the natural log of both sides but I don’t understand how that would help? Bc then you would have
3x^2(8x^5+14)^4 divided by (21x^2 +5x-10)^2 No you would not.
Or would multiplying by the natural log give you
3x^2*4(8x^5+14) divided by 2(21x^2+5x-10) No it would not.
Also, on the other side wouldn’t you have y’/y so at the end you would have to multiply the right side by the original equation to get y’ alone?
I went about it the second way because I thought it seemed easier and I got an insanely long answer
(((84x+10(6x(32x^5+56))-(42x^2+10x-20)(6(32x&5+56)+6x(160x^4))+((84x+10(160x^4(3x^2))-(640x^3(3x^2)+(6x(160x^4)))(42x^2+10x-20)))*(exp(3x^2)(8x^5+14^4)/(21x^2+5x-10)^2)
I also had my friend try and he got:
(((exp(3*(x^2)))*(8*(x^5)+14)^4)/((21*(x^2)+5x-10)^2))*(6*x+((160*(x^4)/(8*(x^5)+14))+(84*x+10)/(21*(x^2)+5*x-10)))
As Sir Michael says, remember your log rules:.
.
\(\displaystyle y = \dfrac{e^{(3x^2)} * (8x^5 + 14)^4}{(21x^2 + 5x - 10)^2}\implies ln(y) = ln\left(\dfrac{e^{3x^2} * (8x^5 + 14)^4}{(21x^2 + 5x - 10)^2}\right) = 3x^2 + 4ln\left(8x^5 + 14\right) - 2ln\left(21x^2 + 5x - 10\right).\)
.
Now differentiate both sides
 
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