Find the derivative of f left parenthesis x right parenthesis equals fraction numerator 3 e to the power of 7 x end exponent over denominator 4 x cubed plus 3 x minus 1 end fraction.A. f apostrophe left parenthesis x right parenthesis equals fraction numerator 84 x cubed e to the power of 7 x end exponent minus 36 x squared e to the power of 7 x end exponent plus 63 x e to the power of 7 x end exponent minus 30 e to the power of 7 x end exponent over denominator open parentheses 4 x cubed plus 3 x minus 1 close parentheses squared end fractionB. f apostrophe left parenthesis x right parenthesis equals fraction numerator 84 x cubed e to the power of 7 x end exponent plus 36 x squared e to the power of 7 x end exponent plus 63 x e to the power of 7 x end exponent minus 12 e to the power of 7 x end exponent over denominator open parentheses 4 x cubed plus 3 x minus 1 close parentheses squared end fractionC. f apostrophe left parenthesis x right parenthesis equals fraction numerator 21 e to the power of 7 x end exponent over denominator 12 x squared plus 3 end fractionD. f apostrophe left parenthesis x right parenthesis equals fraction numerator short dash 84 x cubed e to the power of 7 x end exponent plus 36 x squared e to the power of 7 x end exponent minus 63 x e to the power of 7 x end exponent plus 30 e to the power of 7 x end exponent over denominator open parentheses 4 x cubed plus 3 x minus 1 close parentheses squared end fraction
Question
Find the derivative of f left parenthesis x right parenthesis equals fraction numerator 3 e to the power of 7 x end exponent over denominator 4 x cubed plus 3 x minus 1 end fraction.A. f apostrophe left parenthesis x right parenthesis equals fraction numerator 84 x cubed e to the power of 7 x end exponent minus 36 x squared e to the power of 7 x end exponent plus 63 x e to the power of 7 x end exponent minus 30 e to the power of 7 x end exponent over denominator open parentheses 4 x cubed plus 3 x minus 1 close parentheses squared end fractionB. f apostrophe left parenthesis x right parenthesis equals fraction numerator 84 x cubed e to the power of 7 x end exponent plus 36 x squared e to the power of 7 x end exponent plus 63 x e to the power of 7 x end exponent minus 12 e to the power of 7 x end exponent over denominator open parentheses 4 x cubed plus 3 x minus 1 close parentheses squared end fractionC. f apostrophe left parenthesis x right parenthesis equals fraction numerator 21 e to the power of 7 x end exponent over denominator 12 x squared plus 3 end fractionD. f apostrophe left parenthesis x right parenthesis equals fraction numerator short dash 84 x cubed e to the power of 7 x end exponent plus 36 x squared e to the power of 7 x end exponent minus 63 x e to the power of 7 x end exponent plus 30 e to the power of 7 x end exponent over denominator open parentheses 4 x cubed plus 3 x minus 1 close parentheses squared end fraction
Solution
The derivative of a function can be found using the quotient rule, which states that the derivative of a function in the form of f(x) = g(x)/h(x) is f'(x) = (g'(x)h(x) - g(x)h'(x))/[h(x)]^2.
Given the function f(x) = (3e^(7x))/(4x^3 + 3x - 1), we can identify g(x) = 3e^(7x) and h(x) = 4x^3 + 3x - 1.
The derivative of g(x) = 3e^(7x) is g'(x) = 21e^(7x) (using the chain rule for derivatives).
The derivative of h(x) = 4x^3 + 3x - 1 is h'(x) = 12x^2 + 3.
Applying the quotient rule, we get:
f'(x) = (g'(x)h(x) - g(x)h'(x))/[h(x)]^2 = [(21e^(7x))(4x^3 + 3x - 1) - (3e^(7x))(12x^2 + 3)]/[(4x^3 + 3x - 1)^2]
This simplifies to:
f'(x) = [84x^3e^(7x) + 63xe^(7x) - 21e^(7x) - 36x^2e^(7x) - 9e^(7x)]/[(4x^3 + 3x - 1)^2]
This is not exactly the same as any of the options given, but it is closest to option A. However, option A has a few sign errors. The correct derivative should have a negative sign in front of the 36x^2e^(7x) term and the 9e^(7x) term.
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