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You are here: Experts > Science > Mathematics > Differential Equations > differential calculas
Expert: Scott A Wilson - 10/31/2009
Question
QUESTION: Differentiate the below with chain rule
3√((4-5x^2 )^4 )
ANSWER: 3√((4-5x^2 )^4 ) looks like the same as 3(√((4-5x²)^4), and not a cube root.
Now the squareroot of something to the 4th is that same as that something squared.
That is, the function is 3(4-5x²)².
Let f(t) = 3t² and t(x) = 4-5x² gives us df/dt = 6t and dt/dx = -5x.
Therefore (df/dt)(dt/dx) = df/dx = 6t•-5x where t is 4-5x², so the answer is -5x(4-5x²).
There is no waste in trying to do a derivative on a function if the function can be simplified.
---------- FOLLOW-UP ----------
QUESTION: Scot thanks for Immediate response but i have done one mistake it was cube root so can you help me to reslove this question with this new Equation now with chain rule or the same way you have solve it above . It is urgent if you can help me today it will be gr8
Answer
Then it was (4-5x²)^(4/3). This is (f(x))^(4/3) where f(x) = 4-5x².
The derivative is (4/3)•(f(x))^(1/3)•f'(x), and f'(x) = -10x.
Putting the f'(x) out front and multipling th -10 by 4 gives us (-40/3)x•f'(x)•f(x)^(1/3).
Or as you put it, with the x over by the -40, (-40x/3)•f'(x)•(3√f(x)).
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