Question

Find the derivative of the algebraic function. $$ f(x)=x^{4}\left(1-\frac{2}{x+1}\right) $$

Short answer

The derivative of the algebraic function \(f(x)=x^{4}\left(1-\frac{2}{x+1}\right)\) is \(f'(x) = x^{2}(4x-3)\)

Step-by-step solution

Simplify the function

Before jumping into deriving the function, it is crucial to simplify it first. simplify the function \(f(x)=x^{4}\left(1-\frac{2}{x+1}\right)\) to get it in a form that'll be easier to derive. Distribute the \(x^{4}\) term to the equation to get, \(f(x)=x^{4}-2x^{3}\)

Differentiate

Apply the Power Rule which states that the derivative of \(x^{n}\) is \(nx^{n-1}\). Using the power rule, differentiate \(f(x)=x^{4}-2x^{3}\) term-by-term to get the derivative of f, \(f'(x)= 4x^{3}-6x^{2}\)

Simplify the Derived Function

Finally, simplify the derivative function \(f'(x)= 4x^{3}-6x^{2}\), by taking out a common factor of \(x^2\), thus getting \(f'(x) = x^{2}(4x-3)\)