Question

Us Quotient rule to differentiate the function f(x)=3x/(x^(4)+8)

Answer 1

d/dx((3 x)/(x^4+8))
Factor out constants:
= 3 (d/dx(x/(x^4+8)))
Use the quotient rule, d/dx(u/v) = (v ( du)/( dx)-u ( dv)/( dx))/v^2, where u = x and v = x^4+8:
= 3 ((x^4+8) (d/dx(x))-x (d/dx(x^4+8)))/(x^4+8)^2
The derivative of x is 1:
= (3 (1 (x^4+8)-x (d/dx(x^4+8))))/(x^4+8)^2
Differentiate the sum term by term:
= (3 (-x (d/dx(x^4)+d/dx(8))+x^4+8))/(x^4+8)^2
The derivative of 8 is zero:
= (3 (-x (d/dx(x^4)+0)+x^4+8))/(x^4+8)^2
The derivative of x^4 is 4 x^3:
= (3 (x^4-(4 x^3) x+8))/(x^4+8)^2