how do you differentiate ((x^2 + 4x +3)/(sqrt x))?

Question

this is a math calculus question...

Answer 1

f `(x) =
[ √x ( 2x + 4 ) - ( x ² + 4x + 3 )(1/2)x^(-1/2) ] / x
[ 2x (2x + 4) - (x ² + 4x + 3) ] / 2 x ^(3/2)
[ 3 x ² + 4 x - 3 ] / 2 x ^(3/2)

Answer 2

Using the quotient rule:
(d/dx)((x^2 + 4x +3)/(sqrt x))
= [ sqrt(x)(2x + 4) - (x^2 + 4x + 3)(1 / 2sqrt(x)) ] / x
= sqrt(x) [ 2x(2x + 4) - (x^2 + 4x + 3) ] / 2x^2
= sqrt(x) [ 4x^2 + 8x - x^2 - 4x - 3 ] / 2x^2
= sqrt(x) [ 3x^2 + 4x - 3 ] / 2x^2.

Answer 3

D[f(x)/g(x)] = [f'(x)*g(x) - f(x)*g'(x)]/[g(x)^2]

Answer 4

Use the quotient rule.

f(x) = x^2 + 4x + 3
g(x) = x^0.5

f'(x) = 2x+4
g'(x) = 0.5*x^-0.5

Quotient rule:

[f(x)/g(x)]' = (f'g - g'f)/g^2

So, this becomes:

[(2x+4)*x^0.5 - 0.5*(x^-0.5)*(x^2+4x+3)]/(x^0.5)^2

Let's see if we can simplify:

Denominator is simply x. that's easy.
---------------------------------
Numerator is:

[(2x+4)*x - 0.5*(x^2+4x+3)]/x^0.5 (Notice that I multiplied the first term by (x^0.5)/(x^0.5)

(2x^2+4x - 0.5x^2 - 2x -1.5)/x^0.5

(1.5x^2 - 2x - 1.5)/x^0.5

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Remember that this is just the numerator. The denominator is x so the derivative of f/g is:

(1.5x^2 - 2x - 1.5)/x^1.5

Edit: Please check my arithmetic to be sure I don't have a typo.