Question

1. if f is the function given by $f(x)=\frac{4}{x}+5x - 1$, then $f(2)=$

a 4
b 6
c 7
d 11

1. if $y=\frac{ln x}{x}$, then $\frac{dy}{dx}=$

Explanation:

Question 13

Step1: Find the derivative of \(f(x)\)

Use the power rule \((x^n)^\prime = nx^{n - 1}\). For \(f(x)=\frac{4}{x}+5x - 1=4x^{-1}+5x - 1\), then \(f^\prime(x)=4\times(-1)x^{-2}+5\times1x^{0}-0=- \frac{4}{x^{2}}+5\).

Step2: Evaluate \(f^\prime(x)\) at \(x = 2\)

Substitute \(x = 2\) into \(f^\prime(x)\), \(f^\prime(2)=-\frac{4}{2^{2}}+5\).
First calculate \(\frac{4}{2^{2}}=\frac{4}{4} = 1\), then \(f^\prime(2)=-1 + 5=4\).

Step1: Apply the quotient rule

The quotient rule states that if \(y=\frac{u}{v}\), then \(y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}\). Here \(u=\ln x\), \(u^\prime=\frac{1}{x}\), and \(v = x\), \(v^\prime=1\).

Step2: Calculate \(\frac{dy}{dx}\)

\(\frac{dy}{dx}=\frac{\frac{1}{x}\times x-\ln x\times1}{x^{2}}=\frac{1-\ln x}{x^{2}}\)

Answer:

A. 4

Question 14