Question

Suppose f(x) and g(x) are differentiable functions and g(x)=x(f(x−g(x))).

A) g¹(x)=1−g(x)f¹(x)

B) g¹(x)=f(x)+g(x)f¹(x)

C) g¹(x)=1+g(x)f¹(x)

D) g¹(x)=f(x)−g(x)f¹(x)

Answer 1

To find the derivative of g(x), we can use the product rule.

Step-by-step explanation:

To find the derivative of g(x), we can use the product rule. Let's differentiate g(x) with respect to x:

g'(x) = [x'(f(x-g(x))) + x(f'(x-g(x)) - g'(x))] / x^2

Simplifying further:

g'(x) = [f(x-g(x)) + xf'(x-g(x)) - g'(x)] / x

We can rewrite this as:

g'(x) = f(x-g(x))/x + f'(x-g(x)) - g'(x)/x

Therefore, the correct answer is A) g'(x) = 1 - g(x)f'(x).