Find f ' (x) given that

Question

x²+5x

F(x)= ‾‾‾‾‾‾‾‾‾‾

3x³+4

Answer 1

Assuming you meant F(x) = f(x) then use the quotient rule:

Where u = x² + 5x meaning u' = 2x + 5

And v = 3x³ + 4 meaning that v' = 9x² and v² = (3x³ + 4)²

So f '(x) = [(2x + 5)(3x³ + 4) - (x² + 5x)(9x²)]/(3x³ + 4)²

Simplifying:

f '(x) = [(6x⁴ + 15x³ + 8x + 20) - (9x⁴ + 45x³)]/(3x³ + 4)²

f '(x) = (-3x⁴ - 30x³ + 8x + 20)/(3x³ + 4)²

Answer 2

f(x)=g(x)/h(x)

g(x)=x²+5x

h(x)=3x³+4

f'(x)=(g'(x)h(x)-g(x)h'(x))/(h(x))²

g'(x)=2x+5

h'(x)=9x²

Can you fill in the functions and finish simplifying?