Question

How do you differentiate `(x^2 + 8x + 3 )/ sqrtx` using the quotient rule?

Answer 1

`d/dx(f(x))=(3x^2-3)/(2xsqrt(x))`

Explanation:

The given equation is
`f(x)=(x^2+8x+3)/sqrt(x)`

We use the formula for derivative of rational expression
`d/dx(u/v)=(v⋅d/dx(u)-u⋅d/dx(v))/v^2`

`d/dx(f(x))=d/dx((x^2+8x+3)/sqrt(x))=(sqrt(x)⋅d/dx(x^2+8x+3)-(x^2+8x+3)⋅d/dx(sqrt(x)))/(sqrt(x))^2`

`d/dx(f(x))=(sqrt(x)⋅(2x+8)-(x^2+8x+3)⋅(1/(2sqrt(x))))/(sqrt(x))^2`

`d/dx(f(x))=(((2x)⋅(2x+8)-(x^2+8x+3))/(2sqrt(x)))/(sqrt(x))^2`

`d/dx(f(x))=((2x)⋅(2x+8)-(x^2+8x+3))/(2xsqrt(x))`

`d/dx(f(x))=(4x^2+8x-x^2-8x-3)/(2xsqrt(x))`

`d/dx(f(x))=(3x^2-3)/(2xsqrt(x))`

God bless....I hope the explanation is useful.