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How do you use the chain rule to differentiate #f(x)=sqrt(4x^3+6x)#?

Answer 1

Emilia Aguilar

#f'(x) = (6x^2+3)/sqrt(4x^3+6x)#

Differentiate of #x |-> 4x^3+6x# is #4*3x^2+6*1 = 12x^2+6# and differentiate of #x |-> sqrt(x)# is #1/(2sqrt x)#

Hence by applying the formula #(f(g))' = g'*f'(g)#,

#f'(x) = (12x^2+6)*1/(2*sqrt(4x^3+6x)#

#=> f'(x) = (6x^2+3)/sqrt(4x^3+6x)#

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Answer 2

Charles Anctil

#(6x^2+3)/(sqrt(4x^3+6x))#

#d/dx[sqrt(4x^3+6x)]=(d/dx[(4x^4+6x)^(1/2)])(d/dx[4x^3+6x])=(1/2)(4x^3+6x)^(1/2-1)(12x^2+6)=(12x^2+6)/(2sqrt(4x^3+6x))=(6x^2+3)/(sqrt(4x^3+6x))#

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Answer 3

Scarlett Allison

To differentiate f(x) = sqrt(4x^3 + 6x) using the chain rule, follow these steps:

Identify the outer function and the inner function.
- Outer function: sqrt(u), where u = 4x^3 + 6x
- Inner function: u = 4x^3 + 6x
Calculate the derivative of the outer function with respect to its argument (u) and the derivative of the inner function with respect to x.
Apply the chain rule formula: d/dx[sqrt(u)] = (1/2) * (u)^(-1/2) * du/dx.
Substitute u = 4x^3 + 6x and du/dx = d/dx[4x^3 + 6x] into the chain rule formula to find the derivative of f(x).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.