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

Question

Christopher Allers · Accepted Answer

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

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.

William Abdalla · Answer

undefined To differentiate (x^2 + 8x + 3) / sqrt(x) using the quotient rule:

1. Identify f(x) as the numerator (x^2 + 8x + 3) and g(x) as the denominator (sqrt(x)).
2. Apply the quotient rule: (f'(x)g(x) - f(x)g'(x)) / [g(x)]^2.
3. Find f'(x) and g'(x) by differentiating the numerator and denominator separately.
4. Substitute the values into the quotient rule formula.
5. Simplify the expression to get the final result.

Here's the step-by-step calculation:

f(x) = x^2 + 8x + 3
g(x) = sqrt(x)

f'(x) = 2x + 8
g'(x) = (1/2) * x^(-1/2)

Applying the quotient rule:

[(2x + 8) * sqrt(x) - (x^2 + 8x + 3) * (1/2) * x^(-1/2)] / [sqrt(x)]^2

Simplify the expression:

(2x * sqrt(x) + 8 * sqrt(x) - (x^2 + 8x + 3) / (2 * sqrt(x))) / x

The final result is:

(2x * sqrt(x) + 8 * sqrt(x) - (x^2 + 8x + 3) / (2 * sqrt(x))) / x