Question

given the function ( f(x)=7x^{2}-7ln x ), find ( f(x) ).
answer
( f(x)= )

Explanation:

Step1: Differentiate \(7x^{2}\)

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(y = 7x^{2}\), we have \(y^\prime=7\times2x^{2 - 1}=14x\).

Step2: Differentiate \(-7\ln x\)

Using the formula \((\ln x)^\prime=\frac{1}{x}\), for \(y=-7\ln x\), we have \(y^\prime=-7\times\frac{1}{x}=-\frac{7}{x}\).

Step3: Combine the derivatives

By the sum - difference rule \((u\pm v)^\prime = u^\prime\pm v^\prime\), where \(u = 7x^{2}\) and \(v = 7\ln x\), \(f^\prime(x)=(7x^{2})^\prime-(7\ln x)^\prime\).

Answer:

\(f^\prime(x)=14x-\frac{7}{x}\)