Question

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

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