`F(v) = (v/(v^3 + 1))^6` Find the derivative of the function.

Textbook Question

Chapter 3, 3.4 - Problem 30 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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`F(v)=(v/(v^3+1))^6`

`F(v)=v^6(v^3+1)^-6`

By applying product rule of derivative,

`F'(v)=v^6 d/(dv) (v^3+1)^-6 + (v^3+1)^-6 d/(dv) v^6`

`F'(v)=v^6(-6)(v^3+1)^-7(3v^2) + (v^3+1)^-6 (6v^5)`

`F'(v)=-18v^8(v^3+1)^-7 + 6v^5(v^3+1)^-6`

`F'(v)=(-18v^8)/(v^3+1)^7 + (6v^5)/(v^3+1)^6`

`F'(v)=(-18v^8+6v^5(v^3+1))/(v^3+1)^7`

`F'(v)=(-18v^8+6v^8+6v^5)/(v^3+1)^7`

`F'(v)=(-12v^8+6v^5)/(v^3+1)^7`

`F'(v)=(6v^5(1-2v^3))/(v^3+1)^7`

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hkj1385 | (Level 1) Assistant Educator

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Note:- 1) If y = x^n ; then dy/dx = n*{x^(n-1)}

2) If a function to be differentiated contains sub-functions,then by the rule of differentiation, the last function is differentiated first.

3) If the function is of the form y = u/v ; where u & v are both functions of 'x' , then dy/dx = y' = [{v*u' - u*v'}/(v^2)]

Now, for the given question , find the solution in the attachment

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