Letg(x)=x−−√(f(x)−1)𝑔(𝑥)=𝑥(𝑓(𝑥)−1). Determine the value ofg′(4)𝑔′(4)iff(4)=2𝑓(4)=2andf′(4)=1𝑓′(4)=1.Select one:a.g′(4)=214𝑔′(4)=214b.g′(4)=−214𝑔′(4)=−214c.g(4)=214𝑔(4)=214d.g′(4)=234
Question
Letg(x)=x−−√(f(x)−1)𝑔(𝑥)=𝑥(𝑓(𝑥)−1). Determine the value ofg′(4)𝑔′(4)iff(4)=2𝑓(4)=2andf′(4)=1𝑓′(4)=1.Select one:a.g′(4)=214𝑔′(4)=214b.g′(4)=−214𝑔′(4)=−214c.g(4)=214𝑔(4)=214d.g′(4)=234
Solution
The function g(x) is given as g(x) = x(f(x) - 1). To find g′(4), we need to take the derivative of g(x) using the product rule. The product rule states that the derivative of two functions multiplied together is the first function times the derivative of the second function plus the second function times the derivative of the first function.
So, the derivative of g(x) is g′(x) = 1*(f(x) - 1) + x*f′(x).
Substituting the given values f(4) = 2 and f′(4) = 1 into the derivative, we get:
g′(4) = 1*(2 - 1) + 4*1 = 1 + 4 = 5.
However, none of the provided options match this result. There might be a mistake in the question or the provided options.
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