Determine unsimplified and simplified derivatives of the following.

Question

Determine unsimplified derivatives:

1. y=3x^2 + 3√x - 5/x^2
2. y=(x^2-x)(√x)
3. y=-6/4√3x

Determine simplified derivative:

1. y=(2x^2-1)^4 (3x^2+1)^2

Answer 1

For each term, we must use the power rule d/dx(xⁿ)=nx^n-1

1. y=3x^2 + 3√x - 5/x^2 = 3x²+ 3x^1/2 -5x^-2 => dy/dx=6x + 1.5x^-1/2+10x^-3
2. y=(x^2-x)(√x) = x^3/2-x^1/2 => dy/dx=(3/2)x^1/2+(1/2)x^-1/2
3. y=-6/4√3x =(-6/4√3)x^-1 => dy/dx=(3/2√3)x^-2

For the last we need the product rule: d/dx[f(x)g(x)]=f'(x)g(x)+g'(x)f(x)

y=(2x^2-1)^4 (3x^2+1)^2

f(x) = (2x²-1)⁴ f'(x)=4(2x²-1)³ * 4x

g(x) = (3x²+1)² g'(x)=2(2x²-1) * 6x

dy/dx = (2x²-1)⁴ * 12x(2x²-1) + 8x(2x²-1)³ * (3x²+1)²