Evaluate f'(x) where (i) f(x)=(2x+8)^(6) ln(3x^2-5x),(ii) f(x)=(ln(4x^2)/(sin4x), (iii) e^(3x+2) cos(4x-6).

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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1) You need to use product rule and chain rule to differentiate the given function with respect to x such that:

`f'(x) = ((2x+8)^6)'* ln(3x^2-5x) + (2x+8)^6* (ln(3x^2-5x))'`

`f'(x) = 6(2x+8)^5*2*ln(3x^2-5x) + (2x+8)^6*((3x^2-5x)')/(3x^2-5x)`

`f'(x) = 12(2x+8)^5*ln(3x^2-5x) + (2x+8)^6*(6x-5)/(3x^2-5x)`

Factoring out `(2x+8)^5`  yields:

`f'(x) = (2x+8)^5*(12*ln(3x^2-5x) + ((2x+8)(6x-5))/(3x^2-5x))`

2) You need to use quotient rule and chain rule to differentiate the given function with respect to x such that:

`f'(x) = ((ln(4x^2))'(sin4x) -(ln(4x^2))(sin4x)')/((sin 4x)^2)`

`f'(x) = ((8x/(4x^2))(sin4x) - 4(ln(4x^2))(cos 4x))/((sin 4x)^2)`

`f'(x) = (2(sin 4x)/x - 4(ln(4x^2))(cos 4x))/((sin 4x)^2)`

3) You need to use quotient rule and chain rule to differentiate the given function with respect to x such that:

`f'(x) = ((e^(3x+2))' cos(4x-6) + e^(3x+2)*(cos(4x-6))')`

 `f'(x) = 3e^(3x+2)*cos(4x-6) - 4e^(3x+2)*(sin(4x-6))`

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