13. if f is the function given by $f(x)=\frac{4}{x}+5x - 1$, then $f(2)=$\na 4\nb 6\nc 7\nd 11\n14. if…

Question 13

Explanation:

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).

Answer:

A. 4

Question 14

Explanation:

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:

(\frac{1 - \ln x}{x^{2}})