for the function ( f(x) = 5x^{3}-7x^{2}+7x + 7 ), find ( f(x) ).\n( f(x)=30x - 14 )

Explanation:

Step1: Differentiate (5x^{3})

Using the power rule ((x^n)' = nx^{n - 1}), for (y = 5x^{3}), (y'=5\times3x^{3 - 1}=15x^{2})

Step2: Differentiate (-7x^{2})

Using the power rule, for (y=-7x^{2}), (y'=-7\times2x^{2 - 1}=-14x)

Step3: Differentiate (7x)

Using the power rule, for (y = 7x), (y'=7\times1x^{1 - 1}=7)

Step4: Differentiate the constant term (7)

The derivative of a constant (C) (here (C = 7)) is (0)

Step5: Combine the derivatives

(f'(x)=(5x^{3})'+(-7x^{2})'+(7x)'+(7)') (f'(x)=15x^{2}-14x + 7+0=15x^{2}-14x + 7)

Answer:

(f'(x)=15x^{2}-14x + 7)