differentiate the function

Question

y=2x^2 sqrt -4x^2+5x-7

Answer 1

 

IF the function is y = f(x)*g(x), where f(x) = 2x²  and g(x) = (-4x²+5x-7)^(1/2), THEN:

 

dy/dx  =  y'  = g(x)*f'(x) + g'(x)*f(x)

 

f'(x) = 4x

g'(x) = (1/2)*  (-4x²+5x-7)^(-1/2) * (-8x+5)

 

dy/dx = y' = [(-4x²+5x-7)^(1/2)]*(4x) + [(1/2)*(-8x+5)*(-4x²+5x-7)^(-1/2)]*2x² 

 

Simplified:  dy/dx = y' = 4x*sqrt(-4x²+5x-7) + [ (-8x³+5x²) / sqrt(-4x²+5x-7) ]

 

                                =  4x(-4x²+5x-7)+(-8x³+5x²)

 

                                =  -24x³  + 25x²  - 28x