Question

Differentiate the following functions:

1.y=1/2(3x^2+1)^2

2.y=3+4x-x^2

Find the derivative of the following

1.y=√x^2-2/2x^3+1

2.y=x^2√6x^2/3x

3.y=2x^2√x

Answer 1

$1) y=\frac{1}{2(3x^2+1)^2}\\ y'=\frac{1}{2}(-2)\frac{1}{(3x^2+1)^3}\cdot(6x)=\\ =\frac{-6x}{(3x^2+1)^2}\\ 2) y=3+4x-x^2\\ y'=0+4-2x=4-2x\\ 3) y=\frac{\sqrt{x^2-2}}{2x^3+1}\\ y'=\frac{\frac{1}{2\sqrt{x^2-2}}\cdot 2x\cdot (2x^3+1)- \sqrt{x^2-2}\cdot 6x^2}{(2x^3+1)^2}=\\ =\frac{x\cdot (2x^3+1)- (x^2-2)\cdot 6x^2}{\sqrt{x^2-2}(2x^3+1)^2}=\\ =\frac{-4x^4+12x^2+x}{\sqrt{x^2-2}(2x^3+1)^2}\\ 4) y=\frac{x^2\sqrt{6x^2}}{3x}\\ y=\frac{x\sqrt{6x^2}}{3}=\frac{\sqrt{6}}{3}x^2\\ y'=\frac{\sqrt{6}}{3}2x=\frac{2\sqrt{6}x}{3}\\ 5) y=2x^2\sqrt{x}=2x^{\frac{5}{2}}\\ y'=2\cdot\frac{5}{2}\cdot x^{\frac{3}{2}}=5x^{\frac{3}{2}}$